NRLF 


B   M   252   SE3 


i&fjiH&BEiKn 


THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

IN  MEMORY  OF 

Professor  Richard  Fayram 
1920-1956 


THE   FOUNDATIONS   OF 
EINSTEIN'S  THEORY  OF  GRAVITATION 


THE  FOUNDATIONS  OF 

EINSTEIN'S  THEORY  OF 

GRAVITATION 

BY 

ERWIN    FREUNDLIGH 

DIRECTOR  OF  THE  EINSTEIN  TOWER 
WITH   A   PREFACE   BY 

ALBERT   EINSTEIN 


TRANSLATED  FROM   THE  FOURTH   GERMAN    EDITION,   WITH    TWO    ESSAYS,   BY 

HENRY    L.    BROSE 

CHRIST  CHURCH,  OXFORD 


WITH   AN   INTRODUCTION   BY 

H.    H.    TURNER,    D.Sc.,    F.R.S. 

SAVILIAN   PROFESSOR  OF  ASTRONOMY  IN  THE  UNIVERSITY  OF  OXFORD 
WITH   FIVE   DIAGRAMS 


NEW   YORK 

E.    P.    DUTTON   AND    COMPANY 
PUBLISHERS 


PRINTED    IN    GREAT   BRITAIN 


Add    to    Lib. 


PREFACE 

DR.  FREUNDLICH  has  undertaken  in 
the  following  essay  to  illumine  the 
ideas  and  observations  which  gave 
rise  to  the  general  theory  of  relativity  so  as 
to  make  them  available  to  a  wider  circle  of 
readers. 

I  have  gained  the  impression  in  perusing 
these  pages  that  the  author  has  succeeded  in 
rendering  the  fundamental  ideas  of  the  theory 
accessible  to  all  who  are  to  some  extent  con- 
versant with  the  methods  of  reasoning  of  the 
exact  sciences.  The  relations  of  the  problem 
to  mathematics,  to  the  theory  of  knowledge, 
physics  and  astronomy  are  expounded  in  a 
fascinating  style,  and  the  depth  of  thought  of 
Riemann,  a  mathematician  so  far  in  advance 
of  his  time,  has  in  particular  received  warm 
appreciation. 

Dr.  Freundlich  is  not  only  highly  qualified 
as  a  specialist  in  the  various  branches  of  know- 
ledge involved  to  demonstrate  the  subject ;  he 
is  also  the  first  amongst  fellow-scientists  who 
has  taken  pains  to  put  the  theory  to  the  test. 

May  his  booklet  prove  a  source  of  pleasure 
to  many  ! 

A.  EINSTEIN 


INTRODUCTION 

THE  Universe  is  limited  by  the  properties  of 
light.  Until  half  a  century  ago  it  was 
strictly  true  that  we  depended  upon  our 
eyes  for  all  our  knowledge  of  the  universe,  which 
extended  no  further  than  we  could  see.  Even  the 
invention  of  the  telescope  did  not  disturb  this 
proposition,  but  it  is  otherwise  with  the  invention 
of  the  photographic  plate.  It  is  now  conceivable 
that  a  blind  man,  by  taking  photographs  and  ren- 
dering their  records  in  some  way  decipherable  by 
his  fingers,  could  investigate  the  universe ;  but 
still  it  would  remain  true,  that  all  his  knowledge  of 
anything  outside  the  earth  would  be  derived  from  the 
use  of  light  and  would  therefore  be  limited  by  its 
properties.  On  this  little  earth  there  is,  indeed, 
a  tiny  corner  of  the  universe  accessible  to  other 
senses  :  but  feeling  and  taste  act  only  at  those 
minute  distances  which  separate  particles  of  matter 
when  "  in  contact :  "  smell  ranges  over,  at  the 
utmost,  a  mile  or  two  ;  and  the  greatest  distance 
which  sound  is  ever  known  to  have  travelled  (when 
Krakatoa  exploded  in  1883)  is  but  a  few  thousand 
miles — a  mere  fraction  of  the  earth's  girdle.  The 
scale  of  phenomena  manifested  through  agencies 
other  than  light  is  so  small  that  we  are  unlikely 
to  reach  any  noteworthy  precision  by  their  study. 
Few  people  who  are  not  astronomers  have  spent 
much  thought  on  the  limitations  introduced  by  the 

vii 


viii  THEORY  OF  GRAVITATION 

news  agency  to  which  we  are  so  profoundly  indebted. 
Light  comes  speedily  but  has  far  to  travel,  and  some 
of  the  news  is  thousands  of  years  old  before  we  get 
it.  Hence  our  universe  is  not  co-existent :  the  part 
close  around  us  belongs  to  the  peaceful  present, 
but  the  nearest  star  is  still  in  the  midst  of  the  late 
War,  for  our  news  of  him  is  three  years  old  ;  other 
stars  are  Elizabethan,  others  belong  to  the  time  of 
the  Pharaohs  ;  and  we  have  alongside  our  modern 
civilization  yet  others  of  prehistoric  date.  The 
electric  telegraph  has  accustomed  us  to  a  world 
in  which  the  news  is  approximately  of  even  date  : 
but  our  forefathers  must  have  been  better  able, 
from  their  daily  experience  of  getting  news  many 
months  old,  to  realize  the  unequal  age  of  the  universe 
we  know.  Nowadays  the  inequality  is  almost 
entirely  the  concern  of  the  astronomer,  and  even  he 
often  neglects  or  forgets  it.  But  when  fundamental 
issues  are  at  stake,  the  time  taken  by  the  messenger 
is  an  essential  part  of  the  discussion,  and  we  must 
be  careful  to  take  account  of  it,  with  the  utmost 
precision. 

Our  knowledge  that  light  had  a  finite  velocity 
followed  on  the  invention  of  the  telescope  and  the 
discovery  of  Jupiter's  satellites  :  the  news  of  their 
eclipses  came  late  at  times  and  these  times  were 
identified  as  those  when  Jupiter  was  unusually  far 
away  from  us.  But  the  full  consequences  of  the 
discovery  were  not  realized  at  first.  One  such 
consequence  is  that  the  stars  are  not  seen  in  their 
true  places,  that  is  in  the  places  which  they  truly 
held  when  the  light  left  them  (for  what  may  have 
happened  to  them  since  we  do  not  know  at  all — they 
may  have  gone  out  or  exploded).  Our  earth  is 
only  moving  slowly  compared  with  the  great  haste 
of  light :  but  still  she  is  moving,  and  consequently 
there  is  "  aberration  " — a  displacement  due  to  the 


INTRODUCTION  ix 

ratio  of  the  two  velocities,  easy  enough  to  recognize 
now,  but  so  difficult  to  apprehend  for  the  first  time 
that  Bradley  spent  two  years  in  worrying  over  the 
conundrum  presented  by  his  observations  before  he 
thought  of  the  solution.  It  came  to  him  unexpect- 
edly, as  often  happens  in  such  cases.  In  his  own 
words — "  at  last  when  he  despaired  of  being  able 
to  account  for  the  phenomena  which  he  had  ob- 
served, a  satisfactory  explanation  of  them  occurred 
to  him  all  at  once  when  he  was  not  in  search  of  it." 
He  accompanied  a  pleasure  party  in  a  sail  upon  the 
river  Thames.  The  boat  in  which  they  were  was 
provided  with  a  mast  which  had  a  vane  at  the  top 
of  it.  It  blew  a  moderate  wind,  and  the  party 
sailed  up  and  down  the  river  for  a  considerable 
time.  Dr.  Bradley  remarked  that  every  time  the 
boat  put  about,  the  vane  at  the  top  of  the  boat's 
mast  shifted  a  little,  as  if  there  had  been  a  slight 
change  in  the  direction  of  the  wind.  The  sailors 
told  him  that  this  was  due  to  the  change  in  the 
boat,  not  the  wind  :  and  at  once  the  solution  of 
his  problem  was  suggested.  The  earth  running 
hither  and  thither  round  the  sun  resembles  the  boat 
sailing  up  and  down  the  river :  and  the  apparent 
changes  of  wind  correspond  to  the  apparent  changes 
in  direction  of  the  light  of  a  star.  But  now  comes 
a  point  of  detail — does  the  vane  itself  affect  the  wind 
just  round  it  ?  And,  similarly,  does  the  earth 
itself  by  its  movement  affect  the  ether  just  round 
it,  or  the  apparent  direction  of  the  light  waves  ? 
This  question  suggested  the  famous  Michelson  and 
Morley  experiment  (Phil.  Mag.,  Dec.  1887).  It  is 
curious  to  think  that  in  the  little  corner  of  the 
universe  represented  by  the  space  available  in  a 
laboratory  an  experiment  should  be  possible  which 
alters  our  whole  conceptions  of  what  happens  in 
the  profoundest  depths  of  space  known  to  us^but 


x  THEORY  OF  GRAVITATION 

so  it  is.  The  laboratory  experiment  of  Michelson 
and  Morley  was  the  first  step  in  the  great  advance 
recently  made.  It  discredited  the  existence  of  the 
virtual  stream  of  ether  which  is  the  natural  an- 
tithesis to  the  earth's  actual  motion.  It  was, 
indeed,  open  to  question  whether  restrictions  of  a 
laboratory  might  not  be  responsible  for  the  result : 
for  the  ether  stream  might  exist,  but  the  laboratory 
in  which  it  was  hoped  to  detect  it  might  be  in  a 
sheltered  eddy.  When  bodies  move  through  the 
air,  they  encounter  an  apparent  stream  of  opposing 
air,  as  all  motorists  know  :  but  by  using  a  glass 
screen  shelter  from  the  stream  can  be  found.  And 
even  without  such  special  screening,  there  may  be 
shelter.  When  a  pendulum  is  set  swinging  in 
ordinary  air,  it  is  found  from  experiments  on  clocks 
that  it  carries  a  certain  amount  of  air  along  with 
it  in  its  movement,  although  the  portion  carried 
probably  clings  closely  to  the  surface  of  the  pen- 
dulum. A  very  small  insect  placed  in  the  region 
might  be  unable  to  detect  the  streaming  of  the  air 
further  out.  In  a  similar  way  it  seemed  possible 
that  as  the  earth  moved  through  the  ether  such 
tiny  insects  as  the  physicists  in  their  laboratories 
might  be  in  a  part  of  the  ether  carried  along  with 
the  earth,  in  which  they  could  not  detect  the  stream- 
ing outside.  But  another  laboratory  experiment, 
this  time  by  Sir  Oliver  Lodge,  discredited  this  ex- 
planation, and  it  was  then  suggested  as  an  alternative 
that  distances  were  automatically  altered  by  move- 
ment. 

It  may  be  well  to  explain  briefly  the  significance 
of  this  alternative.  The  Michelson-Morley  experi- 
ment depended  on  the  difference  between  travelling 
up  and  down  stream,  and  across  it.  To  use  a  few 
figures  may  be  the  quickest  way  of  making  the  point 
clear.  Suppose  a  very  wide,  perfectly  smooth  stream 


INTRODUCTION  xi 

running  at  3  miles  an  hour,  and  that  oarsmen  are  to 
start  from  a  fixed  point  O  in  midstream,  row  out 
in  any  direction  to  a  distance  of  4  miles  from  O, 
and  back  again  to  the  starting-point  O.  Which  is 
the  best  direction  to  choose  ?  We  shall  probably 
all  agree  that  it  will  be  either  directly  up  and  down 
stream,  or  directly  across  it,  and  we  may  confine 
attention  to  these  two  directions.  First  suppose  an 
oarsman  A  starts  straight  across  stream.  To  keep 
straight  he  must  set  his  boat  at  an  angle  to  the 
stream.  If  he  reaches  his  4  mile  limit  in  an  hour, 
the  stream  has  been  virtually  carrying  him  down 
3  miles  in  a  direction  at  right  angles  to  his  course  : 
and  the  well-known  relation  between  the  sides  of 
a  right-angled  triangle  tells  us  that  he  has  effectively 
pulled  5  miles  in  the  hour.  It  will  take  him  simi- 
larly an  hour  to  come  back,  and  the  total  journey 
will  involve  an  effective  pull  of  10  miles. 

Now  suppose  another  oarsman,  B,  of  equal  skill 
elects  to  row  up  stream.  In  two  hours  he  could 
pull  10  miles  if  there  were  no  stream  ;  but  since 
meantime  the  stream  has  pulled  him  back  6  miles 
by  "  direct  action  "  he  will  have  only  just  reached 
the  4  mile  limit  from  the  start,  and  has  still  his 
return  journey  to  go.  No  doubt  he  will  accomplish 
this  pretty  quickly  with  the  stream  to  help  him, 
but  his  antagonist  has  already  got  home  before  he 
begins  the  return.  We  might  have  let  him  do  his 
quick  journey  down  stream  first,  but  it  is  easy  to 
see  that  this  would  gain  him  no  ultimate  advantage. 

Michelson  and  Morley  sent  two  rays  of  light  on 
two  journeys  similar  to  those  of  the  oarsmen  A  and 
B.  The  stream  was  the  supposed  stream  of  ether 
from  east  to  west  which  should  result  from  the 
earth's  movement  of  rotation  from  west  to  east. 
They  confidently  expected  the  return  of  A  before 
that  of  B,  and  were  quite  taken  aback  to  find  the 


xii  THEORY  OF  GRAVITATION 

two  reaching  the  goal  together.  In  the  aquatic 
analogy  of  which  we  have  made  use,  it  would  no 
doubt  be  suspected  that  B  was  really  the  faster 
oar,  which  might  be  tested  by  interchanging  the 
courses  ;  but  there  are  no  known  differences  in  the 
velocity  of  light  which  would  allow  of  a  parallel 
explanation.  There  was,  however,  the  possibility 
that  the  distances  had  been  marked  wrongly,  and 
this  was  tested  by  interchanging  them,  without 
altering  the  "  dead-heat." 

Now  there  are  several  alternative  explanations  of 
this  result.  One  is  that  the  ether  does  not  itself 
exist,  and  therefore  there  is  no  stream  of  it,  actual 
or  apparent ;  and  it  is  to  this  sweeping  conclusion 
that  modern  reasoning,  following  recent  experiments 
and  observations,  is  tending.  The  possibility  of  sav- 
ing the  ether  by  endowing  it  with  four  dimensions 
instead  of  three  is  scarcely  calculated  to  satisfy 
those  who  believed  (until  recently)  that  we  knew 
more  about  the  ether  than  about  matter  itself.  They 
saved  the  ether  for  a  time  by  an  automatic  shortening 
of  all  bodies  in  the  direction  of  their  movement, 
which  explained  the  dead-heat  puzzle.  With  the 
velocities  used  above,  the  goal  attained  by  B  must 
be  automatically  moved  f  of  a  mile  nearer  the 
starting-point,  so  that  B  only  rows  3^  miles  out  and 
back  instead  of  4  miles.  So  gross  a  piece  of 
cheating  would  enable  B  to  make  his  dead-heat, 
but  could  scarcely  escape  detection.  The  shortening 
of  the  course  required  in  the  case  of  light  is  very 
minute  indeed,  because  the  velocities  of  the  heavenly 
bodies  are  so  small  compared  with  that  of  light. 
If  they  could  be  multiplied  a  thousand  times  we 
might  see  some  curious  things,  but  we  have  no 
actual  experience  to  guide  a  forecast. 

It  is  a  great  triumph  for  Pure  Mathematics  that 
it  should  have  devised  a  forecast  for  us  in  it^own 


INTRODUCTION  xiii 

peculiar  way.  Starting  from  axioms  or  postu- 
lates, Einstein,  by  sheer  mathematical  skill,  making 
full  use  of  the  beautiful  theoretical  apparatus 
inherited  from  his  predecessors,  pointed  ultimately 
to  three  observational  tests,  three  things  which 
must  happen  if  the  axioms  and  postulates  were 
well  founded.  One  of  the  tests — the  movement  of 
the  perihelion  of  Mercury's  orbit — had  already  been 
made  arid  was  awaiting  explanation  as  a  standing 
puzzle.  Another — a  displacement  of  lines  in  the 
spectrum  of  the  sun — is  still  being  made,  the  issue 
being  not  yet  clear. 

The  third  suggestion  was  that  the  rays  of  light 
from  a  star  would  be  bent  on  passing  near  the  sun 
by  a  particular  amount,  and  this  test  has  just 
provided  a  sensational  triumph  for  Einstein.  The 
application  was  particularly  interesting  because  it 
was  not  known  which  of  at  least  three  results  might 
be  attained.  If  light  were  composed  of  material 
particles  as  Newton  suggested,  then  in  passing  the 
sun  they  would  suffer  a  natural  deflection  (the  use 
of  the  adjective  is  an  almost  automatic  consequence 
of  modes  of  thought  which  we  must  now  abandon) 
which  we  may  call  N.  On  Einstein's  theory  the 
deflection  would  be  just  twice  this  amount,  E  =  2N. 
But  it  was  thought  quite  possible  that  the  result 
might  be  neither  N  nor  E  but  zero,  and  Professor 
Eddington  remarked  before  setting  out  on  the  recent 
expedition  that  a  zero  result,  however  disappointing 
immediately,  might  ultimately  turn  out  the  most 
fruitful  of  all.  That  was  less  than  a  year  ago. 
Perhaps  a  few  dates  are  worth  remembering.  Eins- 
tein's theory  was  fully  developed  and  stated  in 
November,  1915,  but  news  of  it  did  not  reach  Eng- 
land (owing  to  the  War)  for  some  months.  In 
1917  the  Astronomer  Royal  pointed  out  the  special 
suitability  of  the  Total  Solar  Eclipse  of  May,  1919, 


xiv  THEORY  OF  GRAVITATION 

as  an  occasion  for  testing  Einstein's  Theory.  Pre- 
parations for  two  Expeditions  were  commenced — 
Mr.  Hinks  described  the  geographical  conditions  on 
the  central  line  in  November,  1917 — but  could 
not  be  fully  in  earnest  until  the  Armistice  of  Novem- 
ber, 1918.  In  November,  1919,  the  entirely  satis- 
factory outcome  was  announced  to  the  Royal 
Society  and  characterized  by  the  President  as 
necessitating  a  veritable  revolution  in  scientific 
thought. 

But  when  Mr.  Brose  brought  me  his  translation 
of  the  pamphlet  in  the  spring  of  1919,  the  issue 
was  still  in  doubt.  He  had  become  deeply  in- 
terested in  the  new  theory  while  interned  in  Germany 
as  a  civilian  prisoner  and  had  there  made  this  trans- 
lation. I  encouraged  him  to  publish  it  and  opened 
negotiations  to  that  end,  but  it  was  not  until  we 
enlisted  the  sympathy  of  Professor  Eddington 
(on  his  return  from  the  Expedition)  and  approached 
the  Cambridge  Press  that  a  feasible  plan  of  publica- 
tion was  found.  Professor  Eddington  would  have 
been  a  far  more  appropriate  introducer ;  and  it  is 
only  in  deference  to  his  own  express  wish  that  I 
have  ventured  to  take  up  the  pen  that  he  would 
have  used  to  much  better  purpose.  One  advantage 
I  reap  from  the  decision  :  I  can  express  the  thanks 
of  Mr.  Brose  and  myself  to  him  for  his  practical 
help,  and  perhaps  I  may  add  those  of  a  far  wider 
circle  for  his  own  able  expositions  of  an  intricate 
theory,  which  have  done  so  much  to  make  it  known 
in  England. 

H.  H.  TURNER 

UNIVERSITY  OBSERVATORY, 

OXFORD. 
November  30,   1919 


CONTENTS 


PAGE 

INTRODUCTION.    By  Professor  H.  H.  Turner,  F.R.S.  .      vii 
BIOGRAPHICAL  NOTE  .......     xvi 

SECT. 

1.  THE  SPECIAL  THEORY  OF  RELATIVITY  AS  A  STEPPING- 

STONE  TO  THE  GENERAL  THEORY  OF  RELATIVITY        3 

2.  TWO  FUNDAMENTAL  POSTULATES  IN   THE   MATHE- 

MATICAL FORMULATION  OF  PHYSICAL  LAWS     .      IQ 

3.  CONCERNING  THE  FULFILMENT  OF  THE  TWO  POSTU- 

LATES         22 

(a)  The  line-element  in  the  three-dimensional 

manifold  of  points  in  space,  expressed  in 
a  form  compatible  with  the  two  postu- 
lates   23 

(b)  The  line-element  in   the  four-dimensional 

manifold  of  space-time  points,  expressed 
in  a  form  compatible  with  the  two  postu- 
lates .......  31 

4.  THE   DIFFICULTIES   IN   THE   PRINCIPLES   OF   CLASSI- 

CAL  MECHANICS 37 

5.  EINSTEIN'S  THEORY  OF  GRAVITATION 

(a)  THE   FUNDAMENTAL   LAW    OF   MOTION    AND 

THE    PRINCIPLE    OF    EQUIVALENCE     OF    THE 
NEW   THEORY 45 

(b)  RETROSPECT 55 

6.  THE     VERIFICATION      OF     THE      NEW      THEORY      BY 

ACTUAL  EXPERIENCE 62 

APPENDIX  : 

Explanatory  notes  and  bibliographical  refer- 
ences  69 

ON  THE  THEORY  OF  RELATIVITY.     By  Henry  L.  Brose     105 

SOME  ASPECTS    OF  RELATIVITY.      THE  THIRD  TEST. 

By  Henry  L.  Brose 131 


xv 


BIOGRAPHICAL   NOTE 

ALBERT  EINSTEIN  was  born  in  March,  1879,  in  the  town 
Ulm,  situated  on  the  banks  of  the  Danube  in  Wiirtem- 
berg,  Germany.  He  attended  school  at  Munich,  where 
he  remained  till  his  sixteenth  year. 

His  university  studies  extended  over  the  period  1896- 
1900  at  Zurich,  Switzerland.  He  became  a  citizen  of 
Zurich  in  1901.  During  the  following  seven  years  he 
filled  the  post  of  engineer  in  the  Patent  Office,  Bern. 
He  accepted  a  call  to  Zurich  as  Professor  Extraordinarius 
in  1910,  which  he,  however,  soon  resigned  in  favour  of  a 
permanent  chair  in  Prague  University.  In  1911  he 
decided  to  accept  a  similar  post  in  Zurich.  Since  1914 
he  has  continued  his  researches  in  Berlin  as  a  member  of 
the  Berlin  Academy  of  Sciences. 

His  most  important  achievements  are : 
1905.    The  Special  Theory  of  Relativity. 

The  discovery  that  all  forms  of  energy  possess 

inertia. 

The  law  underlying  the  Brownian  movement. 
The  Quantum-Law  of  the  emission  and  absorp- 
tion of  light. 
1907.    The  fundamental  notions  of  the  general  theory  of 

relativity. 

1912.     The  recognition  of  the  non-Euclidean  nature  of 
space-determination   and  its  connection  with 
gravitation. 
1915.     Gravitational  field  equations. 

Explanation  of  the  motion  of  Mercury's  peri- 
helion. 


XVI 


THE  FOUNDATIONS  OF  EINSTEIN'S 
THEORY  OF  GRAVITATION 

INTRODUCTION 

'  I  COWARDS  the  end  of  1915  Albert  Einstein 
brought  to  its  conclusion  a  theory  of  gravi- 
tation  on  the  basis  of  a  general  principle  of 
relativity  of  all  motions.  His  object  was  to  create 
not  a  visual  picture  of  the  action  of  an  attractive 
force  between  bodies,  but  rather  a  mechanics  of  the 
motions  of  the  bodies  relative  to  one  another  under 
the  influence  of  inertia  and  gravity.  To  attain  this 
difficult  goal,  it  is  true,  many  time-honoured  views 
had  to  be  sacrificed,  but  as  a  reward  a  standpoint 
was  reached  which  had  long  seemed  the  highest 
aim  of  all  who  had  occupied  their  minds  with  theo- 
retical physics.  The  fact  that  these  sacrifices  are 
demanded  by  the  new  theory  must,  indeed,  inspire 
confidence  in  it.  For  the  unsuccessful  attempts 
that  have  been  made  during  the  last  centuries  to 
fit  the  doctrine  of  gravitation  satisfactorily  into  the 
scheme  of  natural  science  necessarily  lead  to  the 
conclusion  that  this  would  not  be  possible  without 
giving  up  many  deeply-rooted  ideas.  As  a  matter 
of  fact,  Einstein  reverted  to  the  foundation  pillars 
of  mechanics  as  starting-points  on  which  to  build 
his  theory,  and  he  did  not  satisfy  himself  by  merely 
reforming  the  Newtonian  law  in  order  to  establish 
a  link  with  the  more  recent  views. 


2          THE  FOUNDATIONS  OF  EINSTEIN'S 

To  get  at  an  understanding  of  Einstein's  ideas,  we 
must  compare  the  fundamental  point  of  view  adopted 
by  Einstein  with  that  of  classical  mechanics.  We 
then  recognize  that  a  logical  development  leads  from 
"  the  special "  principle  of  relativity  to  the  general 
theory,  and  simultaneously  to  a  theory  of  gravitation. 


THEORY  OF  GRAVITATION 


§1 

THE  "  SPECIAL "  THEORY  OF  RELATIVITY  AS 
A  STEPPING  STONE  TO  THE  "GENERAL" 
THEORY  OF  RELATIVITY 

THE  complete  upheaval  which  we  are  witness- 
ing in  the  world  of  physics  at  the  present 
time  received  its  impulse  from  obstacles  which 
were  encountered  in  the  progress  of  electrodynamics. 
Yet  the  important  point  in  the  later  development  was 
that  an  escape  from  these  difficulties  was  possible  * 
only  by  founding  mechanics  on  a  new  basis. 

The  development  of  electrodynamics  took  place 
essentially  without  being  influenced  by  the  results 
of  mechanics,  and  without  itself  exerting  any 
influence  upon  the  latter,  so  long  as  its  range  of 
investigation  remained  confined  to  the  electro- 
dynamic  phenomena  of  bodies  at  rest.  Only  after 
Maxwell's  equations  had  furnished  a  foundation  for 
these  did  it  become  possible  to  take  up  the  study 
of  the  electrodynamic  phenomena  of  moving  media. 
All  optical  occurrences — and  according  to  Maxwell's 
theory  all  these  also  belong  to  the  sphere  of  electro- 

*  Note. — Most  of  the  objections  to  the  new  development  have,  it 
is  admitted,  been  raised  because  a  branch  of  science  which  was  not 
considered  to  have  a  just  claim  to  deal  with  questions  of  mechanics, 
asserted  the  right  of  exercising  a  far-reaching  influence  upon  the  latter, 
extending  even  to  its  foundation.  If,  however,  we  trace  these  objec- 
tions to  their  source,  we  discover  that  they  are  due  to  a  wish  to  give 
mechanics  the  form  of  a  purely  mathematical  science,  similar  to  geo- 
metry, in  spite  of  the  fact  that  it  is  founded  upon  hypotheses  which 
are  essentially  physical :  up  to  the  present,  certainly,  these  hypotheses 
have  not  been  recognized  to  be  such. 


4        THE  FOUNDATIONS  OF  EINSTEIN'S 

dynamics — take  place  either  between  stellar  bodies 
which  are  in  motion  relatively  to  one  another,  or  upon 
the  earth,  which  revolves  about  the  sun  with  a  velocity 
of  about  30  kilometres  per  second,  and  performs, 
together  with  the  sun,  a  translational  motion  of 
about  the  same  order  of  magnitude  through  the 
region  of  the  stellar  system.  Hence  questions  of 
great  fundamental  importance  at  once  asserted 
themselves.  Does  the  motion  of  a  light-source 
leave  its  trace  on  the  velocity  of  the  light  emitted 
by  it  ?  And  what  is  the  influence  of  the  earth's 
motion  on  the  optical  phenomena  which  occur  on 
its  surface,  for  example,  in  optical  experiments  in 
a  laboratory  ?  An  endeavour  was  therefore  to  be 
made  to  find  a  theory  of  these  phenomena  in 
which  electrodynamic  and  mechanical  effects  occurred 
simultaneously  (vide  Note  i).  Mechanics,  which  had 
long  stood  as  a  structure  complete  in  every  detail, 
had  to  stand  the  test  as  to  whether  it  was  capable 
of  supplying  the  fitting  arguments  for  a  description 
of  such  phenomena.  Thus  the  problem  of  electro- 
dynamic  events  in  the  case  of  moving  matter  became 
at  the  same  time  a  decisive  problem  of  mechanics. 
The  first  outstanding  attempt  to  describe  these 
phenomena  for  moving  bodies  was  made  by  H. 
Hertz.  He  extended  Maxwell's  equations  by  ad- 
ditional terms  so  as  also  to  express  the  influence  of  the 
motion  of  matter  on  electrodynamic  phenomena,  and 
in  his  extensions  he  adopted  the  view,  characteristic 
for  his  theory,  that  the  carrier  of  the  electromagnetic 
field,  the  ether,  everywhere  participates  in  the 
motion  of  matter.  Consequently,  in  his  equations 
the  state  of  motion  of  the  ether,  as  denoting  the 
state  of  the  ether,  occurs  as  well  as  the  electro- 
magnetic field.  As  is  well  known,  Hertz's  extensions 
cannot  be  brought  into  harmony  with  the  results  of 
observation,  for  example,  that  of  Fizeau's  experiment 


THEORY  OF  GRAVITATION  5 

(Note  2),  so  that  they  excite  merely  an  historic 
interest  as  a  land-mark  on  the  road  to  an  electro- 
dynamics of  moving  matter.  Lorentz  was  the 
first  to  derive  from  Maxwell's  theory  fundamental 
electrodynamic  equations  for  moving  matter  which 
were  in  essential  agreement  with  observation.  He, 
indeed,  succeeded  in  this  only  by  renouncing  a 
principle  of  fundamental  importance,  namely,  by 
disallowing  that  Newton's  and  Galilei's  principle 
of  relativity  of  classical  mechanics  also  holds  for 
electrodynamics.  The  practical  success  of  Lorentz's 
theory  at  first  almost  made  us  fail  to  see  this  sacrifice, 
but  then  the  disintegration  set  in  at  this  point 
which  finally  made  the  position  of  classical  mechanics 
untenable.  To  understand  this  development  we 
therefore  require  a  detailed  treatment  of  the  principle 
of  relativity  in  the  fundamental  equations  of  physics. 
The  principle  of  relativity  of  classical  mechanics 
is  understood  to  signify  the  consequence,  which 
arises  out  of  Newton's  equations  of  motion,  that 
two  systems  of  co-ordinates,  moving  with  uniform 
motion  in  a  straight  line  with  respect  to  one  another, 
are  to  be  regarded  as  fully  equivalent  for  the  descrip- 
tion of  events  in  the  domain  of  mechanics.  For 
our  observations  on  the  earth  this  means  that  any 
mechanical  event  on  the  surface  of  the  earth — for 
example,  the  motion  of  a  projected  body — does  not 
become  modified  by  the  circumstance  that  the 
earth  is  not  at  rest,  but,  as  is  approximately  the 
case,  is  moving  rectilinearly  and  uniformly.  Yet  this 
postulate  of  relativity  does  not  fully  characterize 
the  Newtonian  principle  of  relativity,  even  if  it 
expresses  that  experimental  fact  which  constitutes 
the  essence  of  the  principle  of  relativity.  The 
postulate  of  relativity  has  yet  to  be  supplemented 
by  those  formulae  of  transformation  by  means 
of  which  the  observer  is  able  to  transform  the 


6        THE  FOUNDATIONS  OF  EINSTEIN'S 

co-ordinates  x,  y,  z,  i  that  occur  in  Newton's  equations 
of  motion  into  those  of  a  system  of  reference  which 
is  moving  uniformly  and  rectilinearly  with  respect 
to  his  own  and  which  has  the  co-ordinates  x' ,  yr ,  z',  t' . 
Here  the  co-ordinates,  xt  y,  z,  that  occur  in  the  New- 
tonian equations  denote  throughout  the  results  of 
measurement  (obtained  by  means  of  rigid  measuring 
rods  according  to  the  rules  of  Euclidean  geometry), 
of  the  spatial  positions  of  the  bodies  during  the 
event  in  question,  and  the  fourth  co-ordinate  t 
denotes  the  point  of  time  assigned  to  the  same 
event  given  by  the  position  of  the  hands  of  a 
clock  placed  at  the  point  at  which  the  event  occurs. 
Classical  mechanics  now  supplemented  the  postulate 
of  relativity  above  formulated  by  equations  of 
transformation  of  the  form  : 

Xr  =  X  —  vt       y'  =  y      z'  =  Z       t'  —  t 

for  the  cases  in  which  we  are  dealing  with  the 
co-ordinate  relations  of  two  systems  of  reference 
moving  with  the  uniform  velocity  v  in  the  direction 
of  the  #-axis  with  respect  to  each  other.  This 
group  of  so-called  Galilei-transformations  is  dis- 
tinguished, even  in  the  case  in  which  the  direction 
of  motion  makes  any  angle  with  the  co-ordinate  axes, 
by  the  circumstance  that  the  time-co-ordinate  t 
always  becomes  transformed  by  the  identity  t  =  t' 
into  the  time- values  of  the  second  system  of  reference ; 
it  is  in  this  that  the  absolute  character  of  the  time- 
measures  manifests  itself  in  the  classical  theory. 
Newton's  equations  of  mechanics  do  not  alter  their 
form  if  we  substitute  the  co-ordinates  x' t  y',  z',  t' 
in  them  for  x,  y,  z,  t  by  means  of  these  equations 
of  transformation.  So  long  as  we  restrict  ourselves 
to  those  systems  of  reference  among  all  others  that 
emerge  out  of  each  other  as  a  result  of  transforma- 
tions of  the  above  type,  there  is  no  sense  in  talking 


THEORY  OF  GRAVITATION  7 

of  absolute  rest  or  absolute  motion.  For  we  may 
freely  decide  to  regard  either  of  two  systems  moving 
in  such  a  way  as  the  one  that  is  at  rest  or  in  motion. 
According  to  classical  mechanics  it  was,  indeed,  be- 
lieved that  only  the  Galilei- transformations  could 
come  into  question  when  we  were  concerned  with  re- 
ferring equivalent  systems  of  reference  to  each  other 
according  to  the  principle  of  relativity.  This,  how- 
ever, is  not  the  case.  The  recognition  of  the  fact 
that  other  equations  of  transformation  may  come 
into  question  for  this  purpose,  and,  indeed,  may  be 
chosen  to  suit  the  facts  of  observation  which  are 
to  be  accounted  for,  the  recognition  of  this  fact 
is  the  characteristic  feature  of  the  "  special  "  theory 
of  relativity  of  Lorentz-Einstein  which  replaced 
that  of  Galilei-Newton.  Lorentz's  fundamental  equa- 
tions of  the  electrodynamics  of  moving  matter  led 
to  it.  This  system  of  electrodynamics,  which  is 
in  satisfactory  agreement  with  observation,  is  founded, 
in  contradistinction  to  Hertz's  theory,  on  the  view 
of  an  absolutely  rigid  ether  at  rest.  Its  funda- 
mental equations  assume  as  its  favoured  system  the 
co-ordinate  system  that  is  at  rest  in  the  ether. 

These  fundamental  electrodynamical  equations 
of  Lorentz,  however,  change  their  form  if,  in  them, 
we  replace  the  co-ordinates  %,  yy  z,  t  of  a  system 
of  reference,  initially  chosen,  by  the  co-ordinates 
xr,  yr ,  z',  t'  of  a  system  moving  uniformly  and 
rectilinear  with  respect  to  the  former  by  means 
of  the  transformation  relationships.  Must  we  infer 
from  this  that  systems  of  reference  which  are 
moving  uniformly  and  rectilinearly  with  respect  to 
each  other  are  not  equivalent  as  regards  electro- 
dynamic  events,  and  that  there  is  no  relativity 
principle  of  electrodynamics  ?  No,  this  inference 
is  not  necessary,  because,  as  remarked,  the  principle 
of  relativity  of  classical  mechanics  with  its  group 


8        THE  FOUNDATIONS  OF  EINSTEIN'S 

of  equations  of  transformation  does  not  represent 
the  only  possible  way  of  expressing  the  equivalence 
of  systems  of  reference  that  are  moving  uniformly 
and  rectilinearly  with  respect  to  each  other.  As  we 
shall  show  in  the  sequel,  the  same  postulate  of 
relativity  may  be  associated  with  another  group  of 
transformations.  Nor  did  experiment  seem  to  offer 
a  reason  for  answering  the  above  question  in  the 
affirmative.  For  all  attempts  to  prove  by  optical 
experiments  in  our  laboratories  on  the  earth  the 
progressive  motion  of  the  latter  gave  a  negative 
result  (Note  2).  According  to  our  observations 
of  electrodynamic  events  in  the  laboratory  the  earth 
may  be  regarded  equally  well  as  at  rest  or  in  motion  ; 
these  two  assumptions  are  equivalent. 

This  led  to  the  definite  conviction  that  in  fact  a 
principle  of  relativity  holds  for  all  phenomena,  be 
their  character  mechanical  or  electrodynamic.  But 
there  can  be  only  one  such  principle,  and  not  one 
for  mechanics  and  another  for  electrodynamics. 
For  two  such  principles  would  annul  each  other's 
effects  because  we  should  be  able  to  derive  a  favoured 
system  from  them  in  the  case  of  events  in  which 
mechanical  and  electrodynamical  events  occur  in 
conjunction,  and  this  favoured  system  would  allow 
us  to  talk  with  sense  of  absolute  rest  or  motion  with 
regard  to  it. 

The  one  escape  from  this  difficulty  is  that  opened 
up  by  Einstein.  In  place  of  the  relativity  principle 
of  Galilei  and  Newton  we  have  to  set  another  which 
comprehends  the  events  of  mechanics  and  electro- 
dynamics. This  may  be  done,  without  altering  the 
postulate  of  relativity  formulated  above,  by  setting 
up  a  new  group  of  transformations,  which  refer  the 
co-ordinates  of  equivalent  systems  of  reference  to 
one  another.  The  fundamental  equations  of  me- 
chanics must,  certainly,  then  be  remodelled  so  that 


THEORY  OF  GRAVITATION  9 

they  preserve  their  form  when  subjected  to  such  a 
transformation.  Starting-points  for  this  remodel- 
ling were  already  given.  For  it  had  been  found 
empirically  that  Lorentz's  fundamental  equations 
of  electrodynamics  allowed  new  kinds  of  trans- 
formations of  co-ordinates,  namely,  those  of  the 
form 

x  —  vt 

nf* 

X    = 


where  c  =  velocity  of  light  in  vacuo. 

The  new  principle  of  relativity  set  up  by  Einstein 
is  as  follows  :  Systems  that  are  moving  uniformly  and 
rectilinearly  with  respect  to  each  other  are  completely 
equivalent  for  the  description  of  physical  events.  The 
equations  of  transformation  that  allow  us  to  pass  from 
the  co-ordinates  of  one  such  system  to  those  of  another 
possible  system,  however,  are  not  then  (for  the  case 
when  both  systems  are  moving  parallel  to  their 
#-axes  with  the  constant  velocity  v)  : — 

x'  =  x  —  vt,  y'  =  y,  z'  =  z,  t'  =  t 
but 


Thus  the  Galilei-Newton  principle  of  relativity 
of  classical  mechanics  and  the  Lorentz-Einstein 
"  special  "  principle  of  relativity  differ  only  in  the 
form  of  the  equations  of  transformation  that  effect 
the  transition  to  equivalent  systems  of  reference 
(Note  3). 
Moreover,  the  relation  of  these  two  different 


10       THE  FOUNDATIONS  OF  EINSTEIN'S 

transformation  formulae  to  each  other  comes  out 
clearly  in  the  circumstance  that  the  equations  of 
transformation  of  Galilei  and  Newton  may  be 
derived  by  a  simple  passage  to  the  limit  from  the 
new  equations  of  Lorentz  and  Einstein.  For  if  we 
assume  the  velocity  v  of  each  system  with  respect 
to  the  other  to  be  very  small  compared  with  the 

velocity  of  light  c,  so  that  the  quotient  — 2  or  ^respec- 
tively, may  be  neglected  in  comparison  with  the 
remaining  terms — an  admissible  assumption  in  all 
cases  with  which  classical  mechanics  had  so  far  dealt 
— the  Lorentz-Einstein  transformations  pass  over 
into  those  of  Newton  and  Galilei. 

It  immediately  suggests  itself  to  us  to  ask  what 
it  is  that  compels  us  to  give  up  the  principle  of 
relativity  of  classical  mechanics,  that  is,  what  are 
the  physical  assumptions  in  its  equations  of  trans- 
formation that  stand,  in  contradiction  with  ex- 
perience ?  The  answer  is  that  the  principle  of 
relativity  of  Newton  and  Galilei  does  not  account 
for  the  facts  of  experience  that  emerge  from  Fiz- 
eau's  and  the  Michelson-Morley  experiment,  and 
from  which  it  may  be  inferred  that  the  velocity 
of  light  has  the  particular  character  of  a  universal 
constant  in  the  transformation  relationships  of 
the  principle  of  relativity.  In  how  far  this  peculiar 
property  of  the  velocity  of  light  receives  expression 
in  the  new  equations  of  transformation  requires 
the  following  detailed  explanation. 

The  equations  of  transformation  of  the  principle 
of  relativity  of  Galilei  and  Newton  contain  a  hypoth- 
esis (which  had  hitherto  not  been  recognized  as 
such).  For  it  had  been  tacitly  assumed  that  the 
following  assumption  was  fulfilled  quite  naturally  : 
if  an  observer  in  a  co-ordinate  system  S  measure  the 


THEORY  OF  GRAVITATION  11 

velocity  v  of  the  propagation  of  some  effect  or  other, 
for  example,  a  sound  wave,  then  an  observer  in 
another  co-ordinate  system  S'  which  is  moving 
relatively  to  S,  necessarily  obtains  a  different 
measure  for  the  velocity  of  propagation  of  the 
same  action.  This  was  to  hold  for  every  finite 
velocity  v.  Only  infinite  velocity  was  to  be  dis- 
tinguished by  the  singular  property  that  it  was  to 
come  out  in  every  system  independently  of  its  state 
of  motion  as  having  exactly  the  same  value  in  all 
the  measurements,  namely,  the  value  infinity. 

This  hypothesis — for  we  are  here,  of  course, 
dealing  only  with  a  purely  physical  hypothesis — im- 
mediately suggested  itself.  Without  further  test 
there  was  no  support  for  supposing  that  also  a 
finite  velocity,  namely,  the  velocity  of  light,  which 
the  nai've  point  of  view  is  inclined  to  endow  with 
infinitely  great  velocity,  would  manifest  the  same 
singular  property. 

The  fact,  however,  which  the  Michelson-Morley 
experiment  helped  us  to  become  aware  of  was  that 
the  law  of  propagation  for  light  is,  for  the  observer, 
independent  of  any  progressive  motion  of  his 
system  of  reference,  and  has  the  property  of  isotropy 
(that  is,  equivalence  of  all  systems)  (cf.  Note  2), 
so  that  it  immediately  suggests  itself  to  us  that 
the  velocity  of  light  is  to  be  considered  as  having 
the  same  value  for  all  systems  of  reference.  The 
recognition  of  the  fact  thus  arrived  at  was,  without 
doubt,  a  surprise,  but  it  will  appear  less  strange 
to  those  who  bear  in  mind  the  particular  role  of 
the  velocity  of  light  in  the  equations  of  Maxwell, 
the  foundation  of  our  theory  of  matter. 

In  consequence  of  this  peculiarity,  the  velocity 
of  light  occurs  in  the  equations  of  kinematics  as  a 
universal  constant.  To  understand  this  better  we 
pursue  the  following  argument.  Long  before  the 


12       THE  FOUNDATIONS  OF  EINSTEIN'S 

advent  of  the  questions  of  electrodynamic  phenomena 
in  moving  bodies  we  might,  on  grounds  of  principle, 
have  suggested  quite  generally  the  question  :  how 
are  the  co-ordinates  in  two  systems  of  reference 
that  are  moving  uniformly  and  rectilinearly  with 
respect  to  each  other  to  be  referred  to  each 
other  ?  We  should  have  been  able  to  attack  the 
purely  mathematical  problem  with  a  full  consciousness 
of  the  assumptions  contained  in  the  hypotheses,  as 
was  actually  done  later  by  Frank  and  Rothe  (Note  4). 
We  then  arrive  at  equations  of  transformation 
that  are  much  more  general  than  those  written 
down  on  p.  9.  By  taking  into  account  the 
special  conditions  that  nature  manifests  to  us,  for 
example  the  isotropy  of  space,  we  may  derive  from 
them  particular  forms,  the  hypothetical  assumptions 
contained  in  which  come  clearly  to  view.  Now, 
in  these  general  equations  of  transformation  a 
quantity  occurs  that  deserves  special  notice.  There 
are  "  invariants  "  of  these  equations  of  transforma- 
tion, that  is,  quantities  that  preserve  their  value 
even  when  such  a  transformation  is  carried  out. 
Among  these  invariants  there  is  a  velocity.  This 
signifies  the  following :  if  an  effect  propagates 
itself  in  one  system  with  the  velocity  v,  then  in 
general  the  velocity  of  propagation  of  the  same 
effect  in  another  system  is  other  than  v,  if  the 
second  system  is  moving  relatively  to  the  first. 
Only  the  invariant  velocity  preserves  its  value  in 
all  systems,  no  matter  with  what  velocity  they  be 
moving  relatively  to  one  another.  The  value  of 
this  invariant  velocity  enters  as  a  characteristic 
constant  into  the  equations  of  transformation. 
Hence,  if  we  wish  to  find  those  transformation 
relations  that  hold  physically,  we  must  find  out  the 
singular  velocity  that  plays  this  fundamental  part. 
To  determine  it  is  the  task  of  the  experimental 


THEORY  OF  GRAVITATION  13 

physicist.  If  he  sets  up  the  hypothesis  that  a 
finite  velocity  can  never  be  such  an  invariant, 
the  general  equations  of  transformation  degenerate 
into  the  transformation-relationships  of  the  prin- 
ciple of  relativity  of  Galilei  and  Newton.  (This 
hypothesis  was  made,  albeit  unconsciously,  in 
Newtonian  mechanics.)  It  had  to  be  discarded 
after  the  results  of  the  Michelson-Morley  and 
Fizeau's  experiment  had  justified  the  view  that 
the  velocity  of  light  c  plays  the  part  of  an  in- 
variant velocity.  Then  the  general  equations 
of  transformation  degenerate  into  those  of  the 
"  special  "  principle  of  relativity  of  Lorentz  and 
Einstein. 

This  remodelling  of  the  co-ordinate-transforma- 
tions of  the  principle  of  relativity  led  to  discoveries 
of  fundamental  importance,  as,  for  example,  to  the 
surprising  fact  that  the  conception  of  the  "  sim- 
ultaneity "  of  events  at  different  points  of  space, 
the  conception  on  which  all  time-measurements 
are  based,  has  only  a  relative  meaning,  that  is, 
that  two  events  that  are  simultaneous  for  one 
observer  will  not,  in  general,  be  simultaneous 
for  another.*  This  deprived  time- values  of  the 

*  The  assertion,  "At  a  particular  point  of  the  earth  the  sun 
rises  at  5  o'clock  10'  6","  denotes  that  "  the  rising  of  the  sun  at  a  par- 
ticular point  of  the  earth  is  simultaneous  with  the  arrival  of  the 
hands  of  the  clock  at  the  position  5  o'clock  10'  6"  at  that  point  of  the 
earth."  In  short,  the  determination  of  the  point  of  time  for  the 
occurrence  of  an  event  is  the  determination  of  the  simultaneity  of 
happening  of  two  events,  of  which  one  is  the  arrival  of  the  hands  of  a 
clock  at  a  definite  position  at  the  point  of  observation.  The  com- 
parison of  the  points  of  time  at  which  one  and  the  same  event  occurs, 
as  noted  by  several  observers  situated  at  different  points,  requires  a 
convention  concerning  the  times  noted  at  the  different  points.  The 
analysis  of  the  necessary  conventions  led  Einstein  to  the  fundamental 
discovery  that  the  conception  "  simultaneous  "  is  only  relative  in- 
asmuch as  the  relation  of  time-measurements  to  one  another  in 
systems  that  are  moving  relatively  to  one  another  is  dependent  on 
their  state  of  motion.  This  was  the  starting-point  for  the  arguments 
that  led  to  the  enunciation  of  the  "  special  principle  of  relativity." 


14       THE  FOUNDATIONS  OF  EINSTEIN'S 

absolute  character  which  had  previously  been  a  great 
point  of  distinction  between  them  and  space  co- 
ordinates So  much  has  been  written  in  recent 
years  about  this  question  that  we  need  not  treat 
it  in  detail  here. 

The  new  form  of  the  equations  of  transformation 
by  no  means  exhausts  the  whole  effect  of  the  prin- 
ciple of  relativity  upon  classical  mechanics.  The 
change  which  it  brought  about  in  the  conception 
of  mass  was  almost  still  more  marked. 

Newtonian  mechanics  attributes  to  every  body  a 
certain  inertial  mass,  as  a  property  that  is  in  no 
wise  influenced  by  the  physical  conditions  to  which 
the  body  is  subject.  Consequently,  the  Principle  of 
the  Conservation  of  Mass  also  appears  in  classical 
mechanics  as  independent  from  the  Principle  of 
the  Conservation  of  Energy.  The  special  principle 
of  relativity  shed  an  entirely  new  light  on  these 
circumstances  when  it  led  to  the  discovery  that 
energy  also  manifests  inertial  mass,  and  it  hereby 
fused  together  the  two  laws  of  conservation,  that  of 
mass  and  that  of  energy,  to  a  single  principle.  The 
following  circumstance  moves  us  to  adopt  this  new 
view  of  the  conception  of  mass. 

The  equations  of  motion  of  Newtonian  mechanics 
do  not  preserve  their  form  when  new  co-ordinates 
have  been  introduced  with  the  help  of  the  Lorentz- 
Einstein  transformations.  Consequently,  the  funda- 
mental equation  of  mechanics  had  to  be  remodelled 
accordingly.  It  was  then  found  that  Newton's 
Second  Law  of  Motion  :  force  =  mass  x  accel.  can- 
not be  retained,  and  that  the  expression  for  the 
kinetic  energy  of  a  body  may  no  longer  be  furnished 
by  the  simple  expression  %mv2,  which  involves 
the  mass  and  the  velocity.  Both  these  results  are 
consequences  of  the  change  which  we  found  neces- 
sary to  make  in  our  view  of  the  nature  of  the  mass 


THEORY  OF  GRAVITATION  15 

of  matter.  The  new  principle  of  relativity  and 
the  equations  of  electrodynamics  led,  rather,  to  the 
fundamentally  new  discovery  that  inertial  mass  is 
a  property  of  every  kind  of  energy,  and  that  a  point- 
mass,  in  emitting  or  absorbing  energy,  decreases  or 
increases,  respectively,  in  inertial  mass,  as  is  shown 
in  Note  5  for  a  simple  case.  The  new  kinematics 
thereby  disposes  of  the  simple  relation  between  the 
kinetic  energy  of  a  body  and  its  velocity  relatively 
to  the  system  of  reference.  The  simplicity  of  the 
expression  for  the  kinetic  energy  in  Newtonian 
mechanics  rendered  possible  the  revolution  of  the 
energy  of  a  body  into  that  (kinetic)  of  its  motion 
and  of  the  internal  energy  of  the  body,  which  is 
independent  of  the  former.  Let  us  consider,  for 
example,  a  vessel  containing  material  particles,  no 
matter  of  what  kind,  in  motion.  If  we  resolve 
the  velocity  of  each  particle  into  two  components, 
namely,  into  the  velocity,  common  to  all,  of  the 
centre  of  gravity  and  the  accidental  velocity  of  a 
particle  relative  to  the  centre  of  gravity  of  the 
system,  then,  according  to  the  formulae  of  classical 
mechanics,  the  kinetic  energy  divides  up  into  two 
parts  :  one  that  contains  exclusively  the  velocity 
of  the  centre  of  gravity  and  that  represents  the 
usual  expression  for  the  kinetic  energy  of  the  whole 
system  (mass  of  the  vessel  plus  the  mass  of  the 
particles),  and  a  second  component  that  involves 
only  the  inner  velocities  of  the  system.  This 
category  of  internal  energy  is  no  longer  possible  so 
long  as  the  expression  for  the  kinetic  energy  con- 
tains the  velocity  not  merely  as  a  quadratic  factor  ; 
so  we  are  led  to  the  view  that  the  internal  energy 
of  the  body  comes  into  expression  in  the  energy 
due  to  its  progressive  motion,  and,  indeed,  as  an 
increase  in  the  inertial  mass  of  the  body. 

This  discovery  of  the  inertia  of  energy  created 


16       THE  FOUNDATIONS  OF  EINSTEIN'S 

an  entirely  new  starting-point  for  erecting  the  struc- 
ture of  mechanics.  Classical  mechanics  regards 
the  inertial  mass  of  a  body  as  an  absolute,  invariable, 
characteristic  quantity.  The  special  theory  of  rela- 
tivity, it  is  true,  makes  no  direct  mention  of  the 
inertial  mass  associated  with  matter,  but  it  tells 
us  that  every  kind  of  energy  has  also  inertia.  But, 
as  every  kind  of  matter  has  at  all  times  a  probably 
enormous  amount  of  latent  energy,  its  inertia  is 
composed  of  two  components  ;  the  inertia  of  the 
matter  and  the  inertia  of  its  contained  energy, 
which  consequently  alters  with  the  amount  of 
the  energy-content.  This  view  leads  us  naturally 
to  ascribe  the  phenomenon  of  inertia  in  bodies  to 
their  energy-content  altogether. 

Thus,  there  arose  the  important  task  of  absorbing 
these  new  discoveries  concerning  the  nature  of 
inert  mass  into  the  principles  of  mechanics.  A 
difficulty  hereby  arose  which,  in  a  certain  sense, 
pointed  out  the  limits  of  achievement  of  the  special 
theory  of  relativity.  One  of  the  fundamental 
facts  of  mechanics  is  the  equality  of  the  inertial 
and  gravitational  mass  of  a  body.  It  is  on  the 
supposition  that  this  is  true  that  we  determine 
the  mass  of  a  body  by  measuring  its  weight.  The 
weight  of  a  body  is,  however,  only  denned  with 
reference  to  a  gravitational  field  (Note  18)  :  in  our 
case,  with  reference  to  the  earth.  The  idea  of 
inertial  mass  of  a  body  is,  however,  introduced  as 
an  attribute  of  matter  without  any  reference  what- 
soever to  physical  conditions  external  to  the  body. 
How  does  the  mysterious  coincidence  in  the  values 
of  the  inertial  and  gravitational  mass  of  a  body 
come  about  ? 

Nor  does  the  special  theory  of  relativity  provide 
an  answer  to  this  question.  The  special  theory  of 
relativity  does  not  even  preserve  the  equality  in 


THEORY  OF  GRAVITATION  17 

the  values  of  inertia  and  gravitational  mass ;  a 
fact  which  is  to  be  reckoned  amongst  the  most 
firmly  established  facts  in  the  whole  of  physics. 
For,  although  the  special  theory  of  relativity  makes 
allowance  for  an  inertia  of  energy,  it  makes  none 
for  a  gravitation  of  energy.  Consequently,  a  body 
which  absorbs  energy  in  any  way  will  register  a 
gain  of  inertia  but  not  of  weight,  thereby  trans- 
gressing the  principle  of  the  equality  of  inertial  and 
gravitational  mass  ;  for  this  purpose  a  theory  of 
gravitational  phenomena,  a  theory  of  gravitation, 
is  required.  The  special  theory  of  relativity  can, 
therefore,  be  regarded  only  as  a  stepping-stone  to 
a  more  general  principle,  which  orders  gravitational 
phenomena  satisfactorily  into  the  principles  of 
mechanics. 

This  is  the  point  where  Einstein's  researches 
towards  establishing  a  general  theory  of  rela- 
tivity set  in.  He  has  discovered  that,  by  extend- 
ing the  application  of  the  relativity-principle  to 
accelerated  motions,  and  by  introducing  gravi- 
tational phenomena  into  the  consideration  of  the 
fundamental  principles  of  mechanics,  a  new  founda- 
tion for  mechanics  is  made  possible,  in  which  all 
the  difficulties  occurring  up  to  the  present  are  solved. 
Although  this  theory  represents  a  consistent  de- 
velopment of  the  knowledge  gathered  by  means  of 
the  special  theory  of  relativity,  it  is  so  deeply 
rooted  in  the  substructure  of  our  principles  of  know- 
ing, in  their  application  to  physical  phenomena, 
that  it  is  possible  thoroughly  to  grasp  the  new 
theory  only  by  clearly  understanding  its  attitude 
toward  these  guiding  lines  provided  by  the  theory 
of  knowledge. 

I  shall,  therefore,  commence  the  account  of  his 
theory  by  discussing  two  general  postulates,  which 
should  be  fulfilled  by  every  physical  law,  but  neither 


18       THE  FOUNDATIONS  OF  EINSTEIN'S 

of  which  is  satisfied  in  classical  mechanics  :  whereas 
their  strict  fulfilment  is  a  characteristic  feature  of 
the  new  theory.  Here  we  have  thus  a  suitable 
point  of  entry  into  the  essential  outlines  of  the 
general  theory  of  relativity. 


THEORY  OF  GRAVITATION  19 


§   2 


TWO  FUNDAMENTAL  POSTULATES  IN  THE 
MATHEMATICAL  FORMULATION  OF  PHY- 
SICAL LAWS 

NEWTON  had  established  the  simple  and 
fruitful  law  that  two  bodies,  even  when 
they  are  not  visibly  connected  with  one 
another,  as  in  the  case  of  the  heavenly  bodies, 
exert  a  mutual  influence,  attracting  one  another 
with  a  force  directly  proportional  to  the  product 
of  their  masses,  and  inversely  proportional  to  the 
square  of  the  distance  between  them.  But  Huygens 
and  Leibniz  refused  to  acknowledge  the  validity  of 
this  law,  on  the  ground  that  it  did  not  satisfy  a 
fundamental  condition  to  which  every  physical 
law  is  subject,  viz.  that  of  continuity  (continuity 
in  the  transmission  of  force,  action  "  by  contact  " 
in  contradistinction  to  action  "at  a  distance "). 
How  were  two  bodies  to  exert  an  influence  upon  one 
another  without  a  medium  between  them  to  transmit 
the  action  ?  The  demand  for  a  satisfactory  answer 
to  this  question  became,  in  fact,  so  imperative 
that  finally,  in  order  to  satisfy  it,  the  existence  of 
a  substance  which  pervaded  the  whole  of  cosmic 
space  and  permeated  all  matter — the  "  luminiferous 
ether " — was  assumed,  although  this  substance 
seemed  to  be  condemned  to  remain  intangible  and 
invisible  (i.e.  imperceptible  to  the  senses  for  all  time) 
and  had  to  be  endowed  with  all  sorts  of  contradic- 
tory properties.  In  the  course  of  time,  however, 


20       THE  FOUNDATIONS  OF  EINSTEIN'S 

there  arose  in  opposition  to  such  assumptions  the 
more  and  more  definite  demand  that,  in  the  formu- 
lation of  physical  laws,  only  those  things  were  to  be 
regarded  as  being  in  causal  connection  which  were 
capable  of  being  actually  observed  :  a  demand  which 
doubtless  originates  from  the  same  instinct  in  the 
search  for  knowledge  as  that  of  continuity,  and 
which  really  gives  the  law  of  causality  the  true 
character  of  an  empirical  law,  i.e.  one  of  actual 
experience. 

The  consistent  fulfilment  of  these  two  postulates 
combined  together  is,  I  believe,  the  mainspring  of 
Einstein's  method  of  investigation ;  this  imbues 
his  results  with  their  far-reaching  importance  in 
the  construction  of  a  physical  picture  of  the  world. 
In  this  respect  his  endeavours  will  probably  not 
encounter  any  opposition  in  the  matter  of  principle 
on  the  part  of  scientists.  For  both  postulates — (i) 
that  of  continuity  and  (2)  that  of  causal  relationship 
between  only  such  things  as  lie  within  the  realm  of 
observation — are  of  an  inherent  nature,  i.e.  contained 
in  the  very  nature  of  the  problem.  The  only 
question  that  might  be  raised  is  whether  it  is  ex- 
pedient to  abandon  such  useful  working  hypotheses 
as  "  forces  at  a  distance." 

The  principle  of  continuity  requires  that  all 
physical  laws  allow  of  formulation  as  differential 
laws,  i.e.  physical  laws  must  be  expressible  in  a 
form  such  that  the  physical  state  at  any  point  is 
completely  determined  by  that  of  the  point  in  its 
immediate  neighbourhood.  Consequently,  the  dis- 
tances between  points,  which  are  at  finite  distances 
from  one  another,  must  not  occur  in  these  laws, 
but  only  those  between  points  infinitely  near  to 
one  another.  The  law  of  attraction  of  Newton 
given  above,  inasmuch  as  it  involves  "  action  at  a 
distance,"  disobeys  the  first  postulate. 


THEORY  OF  GRAVITATION  21 

The  second  postulate,  that  of  a  stricter  form  of 
expression  for  causality  in  its  occurrence  in  physical 
laws,  is  intimately  connected  with  a  general  theory 
of  relativity  of  motions.  Such  a  general  principle 
of  relativity  requires  that  all  possible  systems  of 
reference  in  nature  be  equivalent  for  the  description 
of  physical  phenomena,  and  hence  it  avoids  the 
introduction  of  the  very  questionable  conception 
of  absolute  space  which,  for  reasons  we  know  (see 
§  4),  could  not  be  circumvented  by  Newtonian 
mechanics.  A  general  theory  of  relativity  would, 
in  excluding  the  fictitious  quantity  "  absolute  space/' 
reduce  the  laws  of  mechanics  to  motions  of  bodies 
relative  to  one  another,  which  are  actually  and 
exclusively  what  we  observe.  Thus,  its  laws  would 
be  founded  on  observed  facts  more  completely 
than  are  those  of  classical  mechanics. 

The  rigorous  application  of  the  principles  of 
continuity  and  relativity  in  their  general  form 
penetrates  deeply  into  the  problem  of  the  mathe- 
matical formulation  of  physical  laws.  It  will, 
therefore,  be  essential  at  the  outset  to  enter  into  a 
consideration  of  the  principles  involved  in  the 
latter  process. 


THE  FOUNDATIONS  OF  EINSTEIN'S 


§3 


CONCERNING  THE  FULFILMENT  OF  THE  TWO 
POSTULATES 

A  PHYSICAL  law  is  clothed  in  mathematical 
language  by  setting  up  a  formula.  This 
comprises,  and  represents  in  the  form  of 
an  equation,  all  measurements  which  numerically 
describe  the  event  in  question.  We  make  use  of 
such  formulae,  not  only  in  cases  in  which  we  have 
the  means  of  checking  the  results  of  our  calculations 
at  any  moment  actually  at  our  disposal,  but  also 
when  the  corresponding  measurements  cannot  really 
be  carried  out  in  practice,  but  have  to  be  imagined, 
i.e.  only  take  place  in  our  minds  :  e.g.  when  we 
speak  of  the  distance  of  the  moon  from  the  earth, 
and  express  it  in  metres,  as  if  it  were  really  possible 
to  measure  it  by  applying  a  metre-rule  end  to 
end. 

By  means  of  this  expedient  of  analysis  we  have 
extended  the  range  of  exact  scientific  research 
far  beyond  the  limits  of  measurement  actually 
accessible  in  practice,  both  in  the  matter  of  im- 
measurably large,  as  well  as  in  that  of  immeasurably 
small,  quantities.  Now,  when  such  a  formula  is 
used  to  describe  an  event,  symbols  occur  in  it 
that  stand  for  those  quantities  which  are,  in  a  certain 
sense,  the  ground  elements  of  the  measurements, 
with  the  help  of  which  we  endeavour  to  grip  the 
event ;  thus,  for  example,  in  the  case  of  all  spatial 
measurements,  symbols  for  the  "  length  "  of  a  rod, 


THEORY  OF  GRAVITATION  23 

the  "  volume  "  of  a  cube,  and  so  forth.  In  creating 
these  ground  elements  of  spatial  elements  we  had 
hitherto  been  led  by  the  idea  of  a  rigid  body  which 
was  to  be  freely  movable  in  space  without  altering 
any  of  its  dimensional  relationships.  By  the  re- 
peated application  of  a  rigid  unit  measure  along 
the  body  to  be  measured  we  obtained  information 
about  its  dimensional  relationships.  This  idea  of 
the  ideal  rigid  measuring  rod,  which  is  only  partially 
realizable  in  practice,  on  account  of  all  sorts  of 
disturbing  influences  such  as  the  expansion  due  to 
heat,  represents  the  fundamental  conception  of  the 
geometry  of  measure. 

The  discovery  of  suitable  mathematical  terms, 
which  can  be  inserted  in  a  formula  as  symbols 
for  definite  physical  magnitudes  of  measurements, 
such  as  e.g.  length  of  a  rod,  volume  of  a  cube,  etc., 
in  order  to  shift  the  responsibility,  as  it  were,  for 
all  further  deductions  upon  analysis,  is  one  of  the 
fundamental  problems  of  theoretical  physics  and 
is  intimately  connected  with  the  two  postulates  enun- 
ciated in  §  2. 

To  realize  this  fully,  we  must  revert  to  the  founda- 
tions of  geometry,  and  analyse  them  from  the 
point  of  view  adopted  by  Helmholtz  in  various 
essays,  and  by  Riemann  in  his  inaugural  disserta- 
tion of  1854  :  "On  the  hypotheses  which  lie  at  the 
bases  of  geometry."  Riemann  points  almost  pro- 
phetically to  the  path  now  taken  by  Einstein. 

(a)  THE  LINE-ELEMENT  IN  THE  THREE-DIMENSIONAL 
MANIFOLD  OF  POINTS  IN  SPACE,  EXPRESSED 
IN  A  FORM  COMPATIBLE  WITH  THE  Two  POS- 
TULATES 

Every  point  in  space  can  be  singly  and  unam- 
biguously defined  by  the  three  numbers  xlt  x2,  x3, 


24       THE  FOUNDATIONS  OF  EINSTEIN'S 

which  may  be  regarded  as  the  co-ordinates  of  a 
rectangular  system  of  co-ordinates,  and  which 
distinguish  it  from  all  other  points  ;  a  continuous 
variation  of  these  three  numbers  enables  us  to 
specify  every  single  point  of  space  in  turn.  The 
assemblage  of  points  in  space  represents,  in  Rie- 
mann's  notation,  "  a  multiply  extended  magnitude  " 
(an  n-iold  manifoldness  or  manifold)  between  the 
single  elements  (points)  of  which  a  continuous 
transition  is  possible.  We  are  familiar  with  diverse 
continuous  manifolds,  e.g.  the  system  of  colours,  of 
tones  and  various  others.  A  feature  which  is 
common  to  all  of  them  is  that,  in  order  to  specify 
a  single  element  out  of  the  entire  manifold  (to  define 
a  particular  point,  a  particular  colour,  or  a  particular 
tone),  a  characteristic  number  of  magnitude-deter- 
minations, i.e.  co-ordinates,  is  required  :  this  char- 
acteristic number  is  called  the  dimensions  of  the 
respective  manifold.  Its  value  is  three  for  space, 
two  for  a  plane,  one  for  a  line.  The  system  of  colours 
is  a  continuous  manifold  of  the  dimension  three, 
corresponding  to  the  three  "  primary "  colours, 
red,  green,  and  violet,  by  mixing  which  in  due 
proportions  every  colour  can  be  produced. 

But  the  assumption  of  continuity  for  the  transition 
from  one  element  to  another  in  the  same  manifold, 
and  the  determination  of  the  dimensions  of  the 
latter,  does  not  give  us  any  information  about 
the  possibility  of  comparing  limited  parts  of  the 
same  manifold  with  one  another,  e.g.  about  the 
possibility  of  comparing  two  tones  with  one  another 
or  two  single  colours  ;  i.e.  nothing  has  yet  been 
stated  about  the  metric  relations  (measure-condi- 
tions) of  the  manifold,  about  the  nature  of  the  scale, 
according  to  which  measurements  can  be  undertaken 
within  the  manifold.  In  order  to  be  able  to  do 
this,  we  must  allow  experience  to  give  us  the  facts 


THEORY  OF  GRAVITATION  25 

from  which  to  establish  the  metric  (measure-)  laws 
which  hold  for  each  particular  manifold  (space- 
points,  colours,  tones)  under  various  physical  con- 
ditions ;  these  metric  laws  will  be  different  accord- 
ing to  the  set  of  empirical  facts  chosen  for  this 
purpose.* 

In  the  case  of  the  manifold  of  space-points,  ex- 
perience has  taught  us  that  finite  rigid  point-systems 
can  be  freely  moved  in  space  without  altering 
their  form  or  dimensions  ;  the  conception  of  "  con- 
gruence "  which  has  been  derived  from  this  fact, 
has  become  a  vital  factor  for  a  measure-determina- 
tion, f  It  sets  us  the  problem  of  building  up  a 
mathematical  expression  from  the  numbers  xlt  xz, 
xs,  and  ylt  y2,  y3>  which  are  assigned  to  two  definite 
points  in  space,  and  which  we  may  imagine  as  the 
end-points  of  a  rigid  measuring  rod,  such  that 
this  expression  may  be  regarded  as  a  measure  of 
the  distance  between  them,  that  is,  as  an  expression 
for  the  length  of  the  rod,  and  may  be  introduced  as 
such  into  the  formulae  expressing  physical  laws. 

The  equations  of  physical  laws,  which — in  order 
to  fulfil  the  conditions  of  continuity — must  be 
differential  laws,  contain  only  the  distances  ds,  of 
infinitely  near  points,  so-called  line-elements.  We 
must,  therefore,  inquire  whether  our  two  postulates 
of  §  2  have  any  influence  upon  the  analytical  ex- 
pression for  the  line-element  ds,  and,  if  so,  which 
expression  for  the  latter  is  compatible  with  both. 
Riemann  demands  of  a  line-element  in  the  first 
place  that  it  can  be  compared  in  respect  to  its  length 
with  every  other  line-element  independently  of  its 
position  and  direction.  This  is  a  distinguishing 
characteristic  of  the  metric  conditions  ("  measure 
relations ")  prevalent  in  space ;  in  practice  it 

*  Vide  Note  2.  t  Vide  Note  3. 


26      THE  FOUNDATIONS  OF  EINSTEIN'S 

denotes  that  the  rods  must  be  freely  movable. 
This  peculiarity  does  not  exist,  for  instance,  in  the 
manifold  of  tones  or  in  that  of  colours  (vide  Note  7). 
Riemann  formulates  this  condition  in  the  words, 
"  that  lines  must  have  a  length  independent  of 
their  position  and  that  every  line  is  to  be  measurable 
by  means  of  any  other."  He  then  discovers  that : 
if  xlf  x2,  x3  and  x±  -f-  dxlt  x2  -f  dx2,  x3  -f  dx3  re- 
spectively denote  two  infinitely  near  points  in  space 
and  if  the  continuously  variable  numbers  xlf  x2,  x3 
are  any  co-ordinates  whatsoever  (not  e.g.  necessarily 
rectilinear),  then  the  square  root  of  an  always 
positive,  integral,  homogeneous  function  of  the 
second  degree  in  the  differentials  dxlf  dx2,  dx3  has 
all  the  properties  *  which  the  line-element,  being 
the  expression  for  the  length  of  an  infinitely  small 
rigid  measuring  rod,  must  exhibit.  We  thus  find 
that 


in  which  the  coefficients  g^  are  continuous  func- 
tions of  the  three  variables  xlt  x2,  x3,  gives  us 
an  expression  for  the  line-element  at  the  point 

lj        2>        3* 

In  this  expression  no  assumptions  are  made 
concerning  the  nature  of  the  co-ordinates  that 
are  represented  by  the  three  variables,  xlf  x2,  x3, 
that  is,  concerning  particular  metrical  properties 
of  the  manifold  that  go  beyond  the  postulate  of 
the  freedom  of  movement  of  the  measuring  rods. 
But,  if  we  demand,  in  particular,  that  each  point 
of  the  manifold  may  be  fixed  by  means  of  rectangular 
Cartesian  co-ordinates,  whereby  particular  assump- 
tions are  made  concerning  the  possible  ways  of 
placing  the  measuring  rods,  then  the  line-element, 

*  Vide  Note  8. 


THEORY  OF  GRAVITATION  27 

expressed  in  these  special  variables,   assumes  the 
form  _ 

ds  =  ^dx2  +  dy2  +  dz*. 


Hitherto  this  expression  has  always  been  intro- 
duced for  the  length  of  the  line-element  in  all 
physical  laws.  It  is  contained  in  the  more  general 
expression  of  Riemann's  line-element  ds  as  the 
special  case 

(=itfi  =  v 

SM*  \=  o,  fJi  =(=  v. 

By  restricting  ourselves  to  this  special  form  of  the 
line-element  we  are  enabled  to  use  the  measure 
laws  of  Euclidean  geometry  in  all  our  space- 
measurements. 

But  this  particular  assumption  concerning  the 
metrical  constitution  of  space  contains  the  hypoth- 
esis, as  Helmholtz  has  shown  in  a  detailed  discussion, 
that  finite  rigid  point-systems,  i.e.  finite  fixed  dis- 
tances, are  capable  of  unrestrained  motion  in  space, 
and  can  be  made  (by  superposition)  to  coincide 
with  other  (congruent)  point-systems.  With  respect 
to  the  postulate  of  continuity,  this  hypothesis  seems 
inconsistent,  in  so  far  as  it  introduces  implicit 
statements  about  finite  distances  into  purely  dif- 
ferential laws,  in  which  only  line-elements  occur  ; 
but  it  does  not  contradict  the  postulate. 

The  postulate  of  the  relativity  of  all  motion 
adopts  a  different  attitude  towards  the  possibility  ef 
giving  the  line-element  the  Euclidean  form  in 
particular.* 

*  Strictly  speaking,  I  should  at  this  juncture  state  in  anticipation 
that  the  above  investigations  can  manifestly  also  be  so  generalized 
as  to  be  valid  for  the  four-dimensional  space-time  manifold,  in  which 
all  events  actually  take  place,  and  that  the  transformation-formulae 
apply  to  the  four  variables  of  this  manifold.  In  these  general  remarks 
the  neglect  of  the  fourth  dimension  is  of  no  importance.  This  state- 
ment will  be  justified  later  in  §  3  (£). 


28       THE  FOUNDATIONS  OF  EINSTEIN'S 

According  to  the  principle  of  the  relativity  of  all 
motions,  all  systems,  which  come  about  owing  to 
relative  motions  of  bodies  towards  one  another,  may 
be  regarded  as  fully  equivalent.  The  laws  of  physics 
must,  therefore,  preserve  their  form  in  passing 
from  one  such  system  to  another  ;  i.e.  the  trans- 
formation-formulae of  the  variables  xlf  x2,  #3  which 
perform  this  transition  to  another  system,  must  not 
alter  the  analytical  expression  for  the  physical 
law  under  consideration. 

This  leads  us  to  set  up  a  principle  of  relativity 
which  will  be  called  the  general  principle  of  relativity 
in  the  sequel.  It  demands  the  invariance  of 
physical  laws  with  respect  to  arbitrary  continuous 
substitutions  of  the  four  variables.  Moreover,  the 
line-element  that  occurs  in  it  must  preserve  its 
form  when  subjected  to  any  arbitrary  transformations 
whatsoever.  This  condition  is  fully  satisfied  by 
the  line-element 


in  which  no  restrictive  reservations  of  any  description 
are  made  as  to  what  the  co-ordinates  xlt  x2,  x3  are 
to  signify.  The  Euclidean  line-element 

ds  =  +/dx*  -f  dy*  +  dz2, 


on  the  other  hand,  preserves  its  form  only  for  trans- 
formations of  the  special  theory  of  relativity,  which 
confine  themselves  to  systems  moving  uniformly 
and  rectilinearly.  Consequently,  the  element  of 
arc  must  be  adapted  to  the  further  requirements 
of  a  general  theory  of  relativity  so  that  it  preserves 
its  form  after  any  substitutions  whatsoever. 

3 

The  choice  of  the  expression  ds2  =  Zg^x^dx,  to 

i 


THEORY  OF  GRAVITATION  29 

represent  the  line-element  in  physical  laws  is,  in 
spite  of  its  very  general  character,  still  to  be  re- 
garded as  a  hypothesis,  as  Riemann  has  already 
pointed  out.  For  there  are  other  functions  of  the 
differentials  dxlf  dxz,  dx3 — such  as  e.g.  the  fourth 
root  of  a  homogeneous  differential  expression  of 
the  fourth  degree  in  these  variables — which  could 
provide  a  measure  for  the  length  of  the  line-element 
(vide  Note  9).  But  at  present  there  is  no  ground 
for  abandoning  the  simplest  general  expression  for 
the  line-element  (viz.  that  of  the  second  degree), 
and  adopting  more  complicated  functions.  Within 
the  range  (of  fulfilment)  of  the  two  postulates, 
which  we  have  imposed  upon  every  description  of 
physical  events,  the  former  expression  for  ds  satisfies 
all  requirements.  Nevertheless,  it  must  never  be 
forgotten  that  the  choice  of  an  analytical  expression 
for  the  line-element  always  contains  a  hypothetical 
factor  ;  and  it  is  the  duty  of  the  physicist  to  remain 
fully  conscious  of  this  fact  at  all  times,  without 
being  in  any  way  prejudiced.  It  is  for  this  reason 
that  Riemann  closes  his  essay  with  the  following 
remarks,  which  impress  one  particularly  with  their 
great  importance  for  the  present  time  :  * 

"  The  question  of  the  validity  of  the  hypotheses  of 
geometry  in  the  infinitely  small  is  bound  up  with 
the  question  of  the  ground  of  the  metric  relations 
of  space.  In  this  question,  which  we  may  still 
regard  as  belonging  to  the  doctrine  of  space,  is 
found  the  application  of  the  remark  made  above  ; 
that  in  a  discrete  f  manifold,  the  principle  or 
character  of  its  metric  relations  is  already  given 
in  the  notion  of  the  manifold,  whereas  in  a 

*  B.  Riemann,  Uber  die  Hypothesen,  welche  der  Geometric 
zugrunde  liegen.  New  edn.,  annotated  by  H.  Weyl,  Berlin  :  Springer 
&  Co.,  1919. 

t  Vide  Note  6. 


30        THE  FOUNDATIONS  OF  EINSTEIN'S 

continuous  manifold  this  ground  has  to  be  found 
elsewhere,  i.e.  has  to  come  from  outside.  Either, 
therefore,  the  reality  which  underlies  space  must 
form  a  discrete  *  manifold,  or  we  must  seek  the 
ground  of  its  metric  relations  (measure-conditions) 
outside  it,  in  binding  fo/ces  which  act  upon  it. 

A  decisive  answer  to  these  questions  can  be 
obtained  only  by  starting  from  the  conception  of 
phenomena  which  has  hitherto  been  justified  by 
experience,  and  of  which  Newton  laid  the  foundation, 
and  then  making  in  this  conception  the  successive 
changes  required  by  facts  which  admit  of  no  ex- 
planation on  the  old  theory ;  researches  of  this 
kind,  which  commence  with  general  notions,  cannot 
be  other  than  useful  in  preventing  the  work  from 
being  hampered  by  too  narrow  views,  and  in  keeping 
progress  in  the  knowledge  of  the  inter-connections 
of  things  from  being  checked  by  traditional  pre- 
judices. 

.  This  carries  us  over  into  the  sphere  of  another 
science,  that  of  physics,  into  which  the  character 
and  purpose  of  the  present  discussion  will  not 
allow  us  to  enter." 

That  is  to  say :  according  to  Riemann's  view 
these  questions  are  to  be  solved  by  starting  from 
Newton's  view  of  physical  phenomena,  and  com- 
pelled by  facts  which  do  not  allow  of  any  explana- 
tion by  it,  gradually  remoulding  it.  This  is  what 
Einstein  has  done.  The  "  binding  forces,"  to 
which  Riemann  points,  will  be  found  again  in 
Einstein's  theory.  As  we  shall  see  in  the  fifth 
chapter,  Einstein's  theory  of  gravitation  is  based 
upon  the  view  that  the  gravitational  forces  are 
the  "  binding  forces,"  i.e.  they  represent  the  "  inner 
ground  "  of  the  metric  conditions  (measure-relations) 
in  space. 

*  Vide  Note  10. 


THEORY  OF  GRAVITATION  31 


(b)  THE  LINE-ELEMENT  IN  THE  FOUR-DIMENSIONAL 
MANIFOLD  OF  SPACE-TIME  POINTS,  EXPRESSED 
IN  A  FORM  COMPATIBLE  WITH  THE  TWO  POST- 
ULATES 

The  measure-conditions,  which  we  were  to  take 
as  a  basis  for  the  formulation  of  physical  laws, 
could  have  been  treated  immediately  in  connection 
with  the  four-dimensional  manifold  of  space-time 
points.  For  the  special  theory  of  relativity  has 
led  us  to  make  the  important  discovery  that  the 
space-time-manifold  has  uniform  measure-relations 
in  its  four  dimensions.  Nevertheless,  I  wish  to 
treat  time-measurements  separately  ;  for  one  reason 
that  it  is  just  this  result  of  the  relativity- theory 
which  has  experienced  the  greatest  opposition  at 
the  hands  of  supporters  of  classical  mechanics ; 
and  for  another  that  classical  mechanics  is  also 
obliged  to  establish  certain  conditions  about  time- 
measurement,  but  that  it  never  succeeded  in  estab- 
lishing agreement  on  this  point.  The  difficulties 
with  which  classical  mechanics  had  to  contend  are 
contained  in  its  fundamental  conceptions.  The 
law  of  inertia,  particularly,  was  a  permanent  factor 
of  discord  that  caused  the  foundations  of  mechanics 
to  be  incessantly  criticised.  And  since  the  founda- 
tions of  time-measurement  had  been  brought  into 
close  relationship  with  the  law  of  inertia,  these 
critical  attacks  applied  to  them  likewise. 

In  Galilei's  law  of  inertia,  a  body  which  is  not 
subject  to  external  influences  continues  to  move 
with  uniform  motion  in  a  straight  line.  Two 
determining  elements  are  lacking,  viz.  the  reference 
of  the  motion  to  a  definite  system  of  co-ordinates, 
and  a  definite  time-measure.  Without  a  time- 
measure  one  cannot  speak  of  a  uniform  velocity. 


32        THE  FOUNDATIONS  OF  EINSTEIN'S 

Following  a  suggestion  by  C.  Neumann,*  the  law 
of  inertia  has  itself  been  adduced  to  give  a  definition 
of  a  time-measure  in  the  form :  "  Two  material 
points,  both  left  to  themselves,  move  in  such  a 
way  that  equal  lengths  of  path  of  the  one  correspond 
to  equal  lengths  of  path  of  the  other."  On  this 
principle,  into  which  time-measure  does  not  enter 
explicitly,  we  can  define  "  equal  intervals  of  time 
as  such,  within  which  a  point,  when  left  to  itself, 
traverses  equal  lengths  of  path." 

This  is  the  attitude  which  was  also  taken  up  by 
L.  Lange,  H.  Seeliger,  and  others,  in  later  researches. 
Maxwell  selected  this  definition  too  (in  "  Matter  and 
Motion").  On  the  other  hand,  H.  Streintzf  (follow 
ing  Poisson  and  d'Alembert)  has  demanded  the 
disconnection  and  independence  of  the  time-measure 
from  the  law  of  inertia,  on  the  ground  that  the 
roots  of  the  time-concept  have  a  deeper  and  more 
general  foundation  than  the  law  of  inertia.  Accord- 
ing to  his  opinion,  every  physical  event,  which  can 
be  made  to  take  place  again  under  exactly  the  same 
conditions,  can  serve  for  the  determination  of  a 
time-measure,  inasmuch  as  every  identical  event 
must  claim  precisely  the  same  duration  of  time  ; 
otherwise,  an  ordered  description  of  physical  events 
would  be  out  of  the  question.  In  point  of  fact, 
the  clock  is  constructed  on  this  principle.  It  is 
this  principle  which  enables  an  observer  to  under- 
take a  time-measurement  at  least  for  his  place  of 
observation.  The  reduction  of  time-measurements 
to  a  dependence  upon  the  law  of  inertia,  on  the  other 
hand,  leads  to  an  unobjectionable  definition  of  equal 
lengths  of  time  ;  but  the  measurement  of  the  equal 
paths  traversed  by  uniformly  moving  bodies,  and 
the  establishment  of  a  unit  of  time  involved  therein, 
are  only  then  possible  for  a  place  of  observation, 

*  Vide  Note  II.  t  Vide  Note  12. 


THEORY  OF  GRAVITATION  33 

when  the  observer  and  the  moving  body  are  in 
constant  connection,  e.g.  by  light-signals.  It  cannot, 
however,  be  straightway  assumed  that  two  observers, 
who  are  in  rectilinear  motion  relatively  to  one 
another,  and,  therefore,  according  to  the  law  of 
inertia,  equivalent  as  reference  systems,  would 
in  this  manner  gain  identical  results  in  their  time- 
measurements.  Poisson's  idea  thus  leads  to  a 
satisfactory  time-measurement  for  a  given  place  of 
observation  itself  ;  i.e.  in  a  certain  sense  it  allows 
the  construction  of  a  clock  for  that  place.  But  it 
does  not  broach  the  question  of  the  time-relations 
of  different  places  with  one  another  at  all ;  whereas 
Neumann's  suggestion  leads  directly  to  those  ques- 
tions which  have  been  a  centre  of  discussion  since 
Einstein's  enunciation  of  the  relativity-principle. 

In  the  endeavour  to  reduce  classical  mechanics 
to  as  small  a  number  of  principles  as  possible,  in 
perfect  agreement  with  one  another,  writers  resorted 
to  ideal-constructions  and  imaginary  experiments. 

Yet  no  one  conceived  the  idea  that  in  fixing  a 
unit  of  time  on  the  basis  of  the  law  of  inertia,  that 
is,  by  measuring  a  length  (the  distance  traversed), 
the  state  of  motion  of  the  observer  might  exert 
an  influence.  It  was  assumed  that  the  data  ob- 
tained from  the  necessary  observations  had  an 
absolute  meaning  quite  independent  of  the  conditions 
of  observation  when  simultaneous  moments  were 
chosen  and  a  length  was  evaluated.  As  Einstein 
has  shown,  however,  this  is  not  the  case.  Rather, 
this  recognition  of  the  relativity  of  space-  and 
time-measurements  formed  the  starting  point  of 
his  principle  of  relativity  (Note  13).  It  is  a  neces- 
sary consequence  of  the  universal  significance  of 
the  velocity  of  light,  of  which  we  spoke  in  the 
first  section.  Its  recognition  furnished  us  at  once 
with  the  correct  formulae  of  transformation,  allowing 
3 


34      THE  FOUNDATIONS  OF  EINSTEIN'S 

us  to  relate  the  space-time  measurements  of  systems 
moving  uniformly  and  rectilinearly  with  respect 
to  each  other,  and  this  is  what  we  are  concerned 
with  in  Neumann's  suggestion  of  fixing  a  measure 
of  time  with  the  aid  of  the  law  of  inertia.  In  the 
new  equations  of  transformation,  t'  is  not  identically 
equal  to  t,  but  rather 


The  time-measurements  in  the  second  system  which 
is  moving  relatively  to  the  first  are  thus  essentially 
conditioned  by  the  velocity  v  of  each  relative  to 
the  other.  Consequently,  the  fixing  of  a  measure 
of  time  on  the  basis  of  the  law  of  inertia,  as  proposed 
by  Neumann,  does  not  at  all  lead  to  the  result 
that  the  time-measurements  are  entirely  indepen- 
dent of  the  state  of  motion  of  the  systems  with 
respect  to  each  other,  as  assumed  in  classical  me- 
chanics. Only  when  the  researches  of  Einstein 
concerning  the  special  theory  of  relativity  had 
been  carried  out,  did  the  fundamental  assumptions 
of  our  time-measurements  become  fully  cleared  up, 
and  thus  a  serious  shortcoming  in  classical  mechanics 
was  made  good. 

That  such  a  fundamental  revision  of  the  assump- 
tions made  regarding  time-measurements  became 
necessary  only  after  so  great  a  lapse  of  time,  is  to 
be  explained  by  the  fact  that  even  the  velocities 
which  occur  in  astronomy  are  so  small,  in  comparison 
with  the  velocity  of  light,  that  no  serious  discre- 
pancies could  arise  between  theory  and  observation. 
So  it  occurred  that  the  weaknesses  of  the  theory — 
in  particular,  those  due  to  the  motional  relations 


THEORY  OF  GRAVITATION  35 

of  various  systems  to  one  another  —  did  not  come 
to  light  until  the  study  of  electronic  motions,  in 
which  velocities  of  the  order  of  that  of  light  occur, 
proved  the  insufficiency  of  the  existing  theory. 

The  details  of  the  effects,  which  result  from  the 
relativity  of  space-time  measurements,  have  so 
frequently  been  discussed  in  recent  years  that  it  is 
only  possible  to  repeat  what  has  already  often 
been  said.  The  essential  point  in  the  discussion  of 
this  section  is  the  recognition  of  the  fact  that 
space  and  time  represent  a  homogeneous  manifold 
of  "  four  "  dimensions,  with  homogeneous  measure- 
relations  (vide  Note  14).  Consequently,  to  be  con- 
sistent, we  must  apply  the  arguments  of  the 
preceding  §  3  (a)  about  the  measure-relations  to  the 
four-dimensional  space-time-manifold  ;  and,  in  view 
of  the  two  fundamental  postulates  (i)  of  continuity 
and  (2)  of  relativity,  and  including  the  time-measure- 
ment as  the  fourth  dimension,  we  must  select 
for  our  line-element  the  expression  : 


ds*  =  gudxS  +  g^xjdxt  +  .  .  .  +g3idxJXi  +  gudxf, 

in  which  the  g^v  (pv  =  i,  2,  3,  4)  are  functions  of 
the  variables  xlt  x2,  xs,  #4. 

Hitherto  we  have  been  led  to  adopt  this  much 
more  general  attitude  towards  the  questions  of  the 
metric  laws  involved  in  physical  formulae  merely 
by  the  desire  not  to  introduce,  from  the  very  outset, 
more  assumptions  into  the  formulations  of  physical 
laws  than  are  compatible  with  both  postulates, 
and  to  bring  about  a  deeper  appreciation  of  the 
points  of  view,  to  which  the  special  theory  of  relativity 
has  led  us. 

We  can  briefly  summarize  by  saying  :  the  adop- 
tion of  Euclidean  metric-conditions  (measure-rela- 
tions) is  compatible  with  the  postulate  of  continuity  ; 
though  the  special  assumptions  thereby  involved 


36      THE  FOUNDATIONS  OF  EINSTEIN'S 

appear  as  restrictive  or  limiting  hypotheses,  which 
need  not  be  made.  But  the  second  postulate, 
the  reduction  of  all  motions  to  relative  motions, 
compels  us  to  abandon  the  Euclidean  measure- 
determination  (cf.  p.  43).  A  description  of  the 
difficulties  still  remaining  in  mechanics  will  make 
this  step  clear. 


THEORY  OF  GRAVITATION  87 


§4 

THE  DIFFICULTIES  IN  THE  PRINCIPLES  OF 
CLASSICAL  MECHANICS 

THE  foundations  of  classical  mechanics  cannot 
be  exhaustively  described  in  a  narrow  space. 
I  can  only  bring  the  unfavourable  side  of 
the   theory  into   prominent   view  for  the   present 
purpose,   without  being  able  to  do  justice  to  its 
great  achievements  in  the  past.     All  doubts  about 
classical  mechanics  set  in  at  the  very  commencement 
with   the   formulation   of   the   law   which   Newton 
places  at  its  head,  the  formulation  of  the  law  of 
inertia. 

As  has  already  been  emphasized  on  page  31,  the 
assertion  that  a  point-mass  which  is  left  to  itself 
moves  with  uniform  velocity  in  a  straight  line, 
omits  all  reference  to  a  definite  co-ordinate  system. 
An  insurmountable  difficulty  here  arises  :  Nature 
gives  us  actually  no  co-ordinate  system,  with 
reference  to  which  a  uniform  rectilinear  motion 
would  be  possible.  For  as  soon  as  we  connect  a 
co-ordinate  system  with  any  body  such  as  the  earth, 
sun,  or  any  other  body — and  this  alone  gives  it  a 
physical  meaning — the  first  condition  of  the  law 
of  inertia  (viz.  freedom  from  external  influences)  is 
no  longer  fulfilled,  on  account  of  the  mutual  gravi- 
tational effects  of  the  bodies.  One  must  accordingly 
either  assign  to  the  motion  of  the  body  a  meaning 
in  itself,  i.e.  grant  the  existence  of  motions  relative 


38      THE  FOUNDATIONS  OF  EINSTEIN'S 

to  "  absolute  "  space,  or  have  recourse  to  mental 
experiments  by  following  the  example  of  C.  Neumann 
and  introducing  a  hypothetical  body  Alpha,  relative 
to  which  a  system  of  axes  is  denned,  and  with 
reference  to  which  the  law  of  inertia  is  to  hold 
(Inertial  system,  vide  Note  15).  The  alternatives 
with  which  one  is  faced  are  highly  unsatisfactory. 
The  introduction  of  absolute  space  gives  rise  to 
the  oft-discussed  conceptual  difficulties  which  have 
gnawed  at  the  foundations  of  Newton's  mechanics. 
The  introduction  of  the  system  of  reference  Alpha 
certainly  takes  the  relativity  of  motions  so  far 
into  account,  that  all  systems  in  uniform  motion 
relative  to  an  Alpha-system  are  established  as 
equivalent  from  the  very  outset,  but  we  can  affirm 
with  certainty  that  there  is  no  such  thing  as  a 
visible  Alpha-system,  and  that  we  shall  never 
succeed  in  arriving  at  a  final  determination  of 
such  a  system.  (It  will,  at  most,  be  possible,  by 
progressively  taking  account  of  the  influences  of 
constellations  upon  the  solar  system  and  upon 
one  another,  to  approximate  to  a  system  of  co- 
ordinates, which  could  play  the  part  of  such  an 
inertial  system  with  a  sufficient  degree  of  accuracy.) 
As  a  result  of  this  objection,  the  founder  of  the  view 
himself,  C.  Neumann,  admits  that  it  will  always  be 
somewhat  unsatisfactory  and  enigmatical,  and  that 
mechanics,  based  on  this  principle,  would  indeed 
be  a  very  peculiar  theory. 

It  therefore  seems  quite  natural  that  E.  Mach 
(vide  Note  16)  should  be  led  to  propose  that  the 
law  of  inertia  be  so  formulated  that  its  relations 
to  the  stellar  bodies  are  directly  apparent.  "  Instead 
of  saying  that  the  direction  and  speed  of  a  rrfass 
u  remains  constant  in  space,  we  can  make  use  of 
the  expression  that  the  mean  acceleration  of  the 
mass  p,  relative  to  the  masses  m,  m',  m"  ...  at 


THEORY  OF  GRAVITATION  39 

distances  r,  rf,  r"  .  .  .,  respectively,  is  zero  or 

d*    Smr  =  Q 


The  latter  expression  is  equivalent  to  the  former 
statement,  as  soon  as  a  sufficient  number  and 
sufficiently  great  and  extensive  masses  are  taken 
into  consideration.  ..."  This  formulation  cannot 
satisfy  us.  For,  in  addition  to  a  certain  requisite 
accuracy,  the  character  of  a  "  contact "  law  is 
lacking,  so  that  its  promotion  to  the  rank  of  a 
fundamental  law  (in  place  of  the  law  of  inertia) 
is  quite  out  of  the  question. 

The  inner  ground  of  these  difficulties  is  without 
doubt  to  be  found  in  an  insufficient  connection  between 
fundamental  principles  and  observation.  As  a  matter 
of  actual  fact,  we  only  observe  the  motions  of  bodies 
relatively  to  one  another,  and  these  are  never  ab- 
solutely rectilinear  nor  uniform.  Pure  inertial  mo- 
tion is  thus  a  conception  deduced  by  abstraction  from 
a  mental  experiment — a  mere  fiction. 

However  necessary  and  fruitful  a  mental  experi- 
ment may  often  be,  there  is  the  ever-present  danger 
that  an  abstraction  which  has  been  carried  unduly 
far  loses  sight  of  the  physical  contents  of  its  under- 
lying notions.  And  so  it  is  in  this  case.  If  there 
is  no  meaning  for  our  understanding  in  talking  of 
the  "  motion  of  a  body  "  in  space,  as  long  as  there 
is  only  this  one  body  present,  is  there  any  meaning 
in  granting  the  body  attributes  such  as  inertial 
mass,  which  arise  only  from  our  observation  of 
several  bodies,  moving  relatively  to  one  another  ? 
If  not,  then  we  cannot  attach  to  the  conception 
"  inertial  mass  of  a  body/'  an  absolute  significance, 
that  is,  a  meaning  which  is  independent  of  all  other 
physical  conditions,  as  has  hitherto  been  done. 


40      THE  FOUNDATIONS  OF  EINSTEIN'S 

Such  doubts  received  fresh  strength  when  the 
special  theory  of  relativity  endowed  every  form  of 
energy  with  inertia  (vide  Note  17). 

The  results  of  the  special  theory  of  relativity 
entirely  unhinged  our  view  of  the  inertia  of  matter, 
for  they  robbed  the  theorem  concerning  the  equality 
of  inertial  and  gravitational  mass  of  its  strict  validity. 
A  body  was  now  to  have  an  inertial  mass  varying 
with  its  contained  internal  energy,  without  its 
gravitational  mass  being  altered.  But  the  mass  of 
a  body  had  always  been  ascertained  from  its  weight, 
without  any  inconsistencies  manifesting  themselves 
(vide  Note  18). 

A  difficulty  of  such  a  fundamental  character  could 
come  to  light  only  owing  to  the  theorem  of  the  equality 
of  inertial  and  gravitational  mass  not  being  sufficiently 
interwoven  with  the  underlying  principles  of  mechanics, 
and  because,  in  the  foundations  of  Newtonian 
mechanics,  the  same  importance  had  not  been  accorded 
to  gravitational  phenomena  as  to  inertial  phenomena, 
which,  judged  from  the  standpoint  of  experience, 
must  be  claimed.  Gravitation,  as  a  force  acting 
at  a  distance,  is,  on  the  contrary,  introduced  only 
as  a  special  force  for  a  limited  range  of  phenomena  : 
and  the  surprising  fact  of  the  equality  of  inertial 
and  gravitational  mass,  valid  at  all  times  and  in 
all  places,  receives  no  further  attention.  One  must, 
therefore,  substitute  for  the  law  of  inertia  a  fundamental 
law  which  comprises  inertial  and  gravitational  phe- 
nomena. This  can  be  brought  about  by  a  consistent 
application  of  the  principle  of  the  relativity  of  all 
motions,  as  Einstein  has  recognized.  This,  is,  there- 
fore, the  circumstance  chosen  by  Einstein  as  a  nucleus 
about  which  to  weave  his  developments. 

The  theorem  of  the  equality  of  inertial  and  gravita- 
tional mass,  which  reflects  the  intimate  connection 
between  inertial  and  gravitational  phenomena,  may 


THEORY  OF  GRAVITATION  41 

be  illuminated  from  another  point  of  view,  and 
thereby  discloses  its  close  relationship  (vide  page  55) 
to  the  general  principle  of  relativity. 

However  much  the  notion  of  "  absolute  space  " 
repelled  Newton,  he  nevertheless  believed  he  had 
a  strong  argument,  in  support  of  the  existence  of 
absolute  space,  in  the  phenomenon  of  centrifugal 
forces.  When  a  body  rotates,  centrifugal  forces 
make  their  appearance.  Their  presence  in  a  body 
alone,  without  any  other  visible  body  being  present, 
enables  one  to  demonstrate  the  fact  that  it  is  in 
rotation.  Even  if  the  earth  were  perpetually 
enveloped  in  an  opaque  sheet  of  cloud,  one  would 
be  able  to  establish  its  daily  rotation  about  its 
axis  by  means  of  Foucault's  pendulum-experiment. 
This  peculiarity  of  rotations  led  Newton  to  conclude 
that  absolute  motions  exist.  From  the  purely 
kinematical  point  of  view,  however,  the  rotation 
of  the  earth  is  not  to  be  distinguished  in  any  way 
from  a  translation  ;  in  this  case,  too,  we  observe 
only  the  relative  motions  of  bodies,  and  might  just 
as  well  imagine  that  all  bodies  in  the  universe 
revolve  around  the  earth.  E.  Mach  has,  in  fact, 
affirmed  that  both  events  are  equivalent,  not  only 
kinematically,  but  also  dynamically :  it  must, 
however,  then  be  assumed  that  the  centrifugal 
forces,  which  are  observed  at  the  surface  of  the  earth, 
would  also  arise,  equal  in  quantity  and  similar  in 
their  manifestations,  from  the  gravitational  effect  of 
all  bodies  in  their  entirety,  if  these  revolved  around 
the  supposedly  fixed  earth  (vide  Note  19). 

The  justification  for  this  view,  which  in  the 
first  place  arises  out  of  the  kinematical  standpoint, 
is,  in  the  main,  to  be  sought  in  the  fact,  derived 
from  experience,  that  inertial  and  gravitational 
mass  are  equal.  According  to  the  conceptions, 
which  have  hitherto  prevailed,  the  centrifugal 


42      THE  FOUNDATIONS  OF  EINSTEIN'S 

forces  are  called  into  play  by  the  inertia  of  the 
rotating  body  (or  rather  by  the  inertia  of  the  separate 
points  of  mass,  which  continually  strive  to  follow 
the  bent  of  their  inertia,  and,  therefore,  express  the 
tendency  to  fly  off  at  a  tangent  to  the  path  in  which 
they  are  constrained  to  move).  The  field  of  centri- 
fugal forces  is,  therefore,  an  inertial  field  (vide 
Note  20).  The  possibility  of  regarding  it  equally 
well  as  a  gravitational  field — and  we  do  that,  as  soon 
as  we  also  assert  the  relativity  of  rotations  dynami- 
cally :  for  we  must  then  assume  that  the  whole 
of  the  masses  describing  paths  about  the  (supposed) 
fixed  body  induce  the  so-called  centrifugal  forces 
by  means  of  their  gravitational  action — is  founded 
on  the  equality  of  inertial  and  gravitational  mass, 
a  fact  which  Eotvos  has  established  with  extra- 
ordinary precision  by  making  use  of  the  centrifugal 
forces  of  the  rotating  earth  (vide  Note  21).  From 
these  considerations  one  realizes  how  a  general  prin- 
ciple of  the  relativity  of  all  motions  simultaneously 
implies  a  theory  of  gravitational  fields. 

From  these  remarks  one  inevitably  gains  the 
impression  that  a  construction  of  mechanics  upon 
an  entirely  new  basis  is  an  absolute  necessity. 
There  is  no  hope  of  a  satisfactory  formulation  of 
the  law  of  inertia  without  taking  into  account  the 
relativity  of  all  motions,  and  hence  just  as  little 
hope  of  banishing  the  unwelcome  conception  of 
absolute  motion  out  of  mechanics  ;  moreover,  the 
discovery  of  the  inertia  of  energy  has  taught  us  facts 
which  refuse  to  fit  into  the  existing  system,  and 
necessitate  a  revision  of  the  foundations  of  mechanics. 
The  condition  which  must  be  imposed  at  the  very 
outset  (cf.  page  20)  is  :  Elimination  of  all  actions 
which  are  supposed  to  take  place  "  at  a  distance  " 
and  of  all  quantities  which  are  not  capable  of  direct 
observation,  out  of  the  fundamental  laws ;  i.e. 


THEORY  OF  GRAVITATION  43 

the  setting-up  of  a  differential  equation  which  com- 
prises the  motion  of  a  body  under  the  influence  of 
both  inertia  and  gravity  and  symbolically  expresses 
the  relativity  of  all  motions.  This  condition  is 
completely  satisfied  by  Einstein's  theory  of  gravita- 
tion and  the  general  theory  of  relativity.  The 
sacrifice,  which  we  have  to  make  in  accepting  them, 
is  to  renounce  the  hypothesis,  which  is  certainly 
deeply  rooted,  that  all  physical  events  take  place 
in  space  whose  measure-relations  (geometry)  are 
given  to  us  a  priori,  independently  of  all  physical 
knowledge.  As  we  shall  see  in  the  following  section, 
the  general  theory  of  relativity  leads  us,  rather, 
to  the  view  that  we  may  regard  the  metrical  condi- 
tions in  the  neighbourhood  of  bodies  as  being  con- 
ditioned by  their  gravitation.  In  this  way  the 
geometry  of  the  measuring  physicist  becomes  in- 
timately welded  with  the  other  branches  of  physics. 
If  we  compress  into  a  short  statement  what  we 
have  so  far  deduced  out  of  the  fundamental  postu- 
lates formulated  at  the  beginning,  we  may  say  : 
The  postulate  of  general  relativity  demands  that 
the  fundamental  laws  be  independent  of  the  par- 
ticular choice  of  the  co-ordinates  of  reference. 
But  the  Euclidean  line-element  does  not  preserve 
its  form  after  any  arbitrary  change  of  the  co- 
ordinates of  reference.  We  have,  therefore,  to 
substitute  in  its  place  the  general  line-element : 

ds*  =  Sg^dx^dXy. 

Whereas,  then,  the  postulate  of  continuity  (cf. 
page  20)  seemed  to  render  it  only  advisable  not  to 
introduce  the  narrowing  assumptions  of  the  Euclidean 
determination  of  measure,  the  principle  of  general 
relativity  no  longer  leaves  us  any  choice. 

The  reason  for  so  emphasizing  the  latter  principle 


44      THE  FOUNDATIONS  OF  EINSTEIN'S 

— as,  indeed,  also  the  postulate  that  only  observable 
quantities  are  to  occur  in  physical  laws — is  not  to 
be  sought  in  any  requirement  of  a  merely  formal 
nature,  but  rather  in  an  endeavour  to  invest  the 
principle  of  causality  with  the  authority  of  a  law 
which  holds  good  in  the  world  of  actual  physical 
experience.  The  postulate  of  the  relativity  of  all 
motions  receives  its  true  value  only  in  the  light  of 
the  theory  of  knowledge  (Note  22).  One  must,  above 
all,  avoid  introducing  into  physical  laws,  side  by 
side  with  observable  quantities,  hypotheses  which 
are  purely  fictitious  in  character,  as  e.g.  the  space 
of  Newton's  mechanics.  Otherwise  the  principle 
of  causality  would  not  give  us  any  real  information 
about  causes  and  effects,  i.e.  the  causal  relations 
of  the  contents  of  direct  experience ;  which  is, 
presumably,  the  aim  of  every  physical  description  of 
natural  phenomena. 


THEORY  OF  GRAVITATION  45 


§5 
EINSTEIN'S  THEORY  OF  GRAVITATION 

(a)  THE  FUNDAMENTAL  LAW  OF  MOTION  AND  THE 
PRINCIPLE  OF  EQUIVALENCE  OF  THE  NEW 
THEORY 

AFTER  the  foregoing  remarks  we  shall  be  able 
to  proceed  to  a  short  account  of  Einstein's 
theory  of  gravitation.  Within  the  limits  of  the 
mathematics  assumed  in  this  book  we  shall,  of  course, 
only  be  able  to  sketch  the  outlines  so  far  that  the  as- 
sumptions and  hypotheses  characteristic  of  the  theory 
come  into  clear  view  and  that  their  relation  to  the 
two  fundamental  postulates  of  the  second  section 
becomes  manifest.  We  start  out  from  the  funda- 
mental law  of  motion  in  classical  mechanics,  the  law 
of  inertia.  Since  even  in  the  law  of  inertia  all 
the  weaknesses  of  the  old  theory  come  to  light, 
a  new  fundamental  law  of  motion  becomes  an  ab- 
solute necessity  for  the  new  mechanics.  It  is 
thus  natural  that  we  should  start  building  up  the 
new  theory  from  this  point.  The  new  law  of  motion 
must  be  a  differential  law,  which,  in  the  first  place, 
describes  the  motion  of  a  point-mass  under  the 
influence  of  both  inertia  and  gravity,  and  which, 
secondly,  always  preserves  the  same  form,  irrespec- 
tive of  the  system  of  co-ordinates  to  which  it  be 
referred,  so  that  no  system  of  co-ordinates  enjoys 
a  preference  to  any  other.  The  first  condition 


46      THE  FOUNDATIONS  OF  EINSTEIN'S 

arises  from  the  necessity  of  ascribing  the  same 
importance  to  gravitational  phenomena  as  to  in- 
ertial  phenomena  in  the  new  process  of  founding 
mechanics — the  law  must,  therefore,  also  contain 
terms  which  denote  the  gravitational  state  of  the 
field  from  point  to  point ;  the  second  condition 
is  derived  from  the  postulate  of  the  relativity  of 
all  motion. 

A  law  of  this  kind  exists  in  the  special  theory  of 
relativity  in  the  equation  of  motion  of  a  single 
point,  not  subject  to  any  external  influence.  Ac- 
cording to  this  equation,  the  path  of  a  point  is  the 
"  shortest  "  or  "  straightest  "  line  (vide  Note  23) 
— i.e.  the  "  straight  line,"  if  the  line-element  ds 
is  Euclidean.  Written  as  an  equation  of  variation 
this  law  is : 


=  8  {J  V  -  dx*  -  dy*  -  dz*  +  cHi*}  =  o. 

If  the  principle  of  the  shortest  path,  which  is  to  be 
followed  in  actual  motions,  be  elevated  in  this 
form  to  a  general  differential  law  for  the  motion 
in  a  gravitational  field  too,  with  due  regard  to  the 
principle  of  the  relativity  of  all  motions,  the  new 
fundamental  law  must  run  as  follows  : 


For  only  this  form  of  the  line-element  remains 
unaltered  (invariant)  for  arbitrary  transformations 
of  the  xlf  xz,  #3,  #4.  The  factors  gn  .  .  .  g44,  which 
for  the  present  we  leave  unexplained,  occur  in  it 
as  something  essentially  new.  Now,  the  extra- 
ordinarily fruitful  idea  that  occurred  to  Einstein 
was  this  :  Since  the  new  law  is  to  hold  for  any 
arbitrary  motions  whatsoever,  thus  also  for  accelera- 
tions, such  as  we  perceive  in  gravitational  fields, 
we  must  make  the  gravitation  field,  in  which 


THEORY  OF  GRAVITATION  47 

observed  motion  takes  place,  responsible  for  the 
occurrence  of  these  ten  factors  g^.  These  ten 
co-efficients  g^  which  will,  in  general,  be  functions 
of  the  variables  xlt  .  .  .  #4,  must,  if  the  new  funda- 
mental law  is  to  be  of  use,  be  able  to  be  brought 
into  such  relationship  to  the  gravitational  field,  in 
which  the  motion  takes  place,  that  they  are  deter- 
mined by  the  field,  and  that  the  motion  described 
by  equation  (i)  coincides  with  the  observed  motion. 
This  is  actually  possible.  (The  g^'s  are  the  gravi- 
tational potentials  of  the  new  theory,  i.e.  they 
take  over  the  part  played  by  the  one  gravitational 
potential  in  Newton's  theory,  without,  however, 
having  the  special  properties,  which  according  to 
our  knowledge  a  potential  has,  in  addition.) 
^Corresponding  to  the  measure-relations  of  a 
space-time  manifold  based  upon  the  line-element : 

4 

ds*  =  2  g^dx^dx,, 
i 

which  is  now  placed  at  the  foundation  of  mechanics 
by  virtue  of  the  relativity  of  all  motions,  the  remain- 
ing physical  laws  must  also  be  so  formulated  that 
they  remain  independent  of  the  accidental  choice 
of  the  variables.  Before  we  enter  into  this  more 
closely,  the  distinguishing  features  of  the  theory 
of  gravitation  characterized  by  equation  (i)  will  be 
considered  in  greater  detail. 

The  postulate  of  the  new  theory,  that  the  laws 
of  mechanics  are  only  to  contain  statements  about 
the  relative  motions  of  bodies,  and  that,  in  particular, 
the  motion  of  a  body  under  the  action  of  the  attrac- 
tion of  the  remaining  bodies  is  to  be  symbolically 
described  by  the  formula : 


=  o, 


48       THE  FOUNDATIONS  OF  EINSTEIN'S 

is  fulfilled  in  Einstein's  theory  by  a  physical  hypoth- 
esis concerning  the  nature  of  gravitational  pheno- 
mena, which  he  calls  the  hypothesis  or  principle 
(respectively)  of  equivalence  (vide  Note  24).  This 
asserts  the  following  : 

Any  change,  which  an  observer  perceives  in  the 
passing  of  any  event  to  be  due  to  a  gravitational  field, 
would  be  perceived  by  him  in  exactly  the  same  way, 
if  the  gravitational  field  were  not  present,  provided 
that  he  —  the  observer  —  makes  his  system  of  reference 
move  with  the  acceleration  which  was  characteristic 
of  the  gravitation  at  his  point  of  observation. 

For,  if  the  variables  x,  y,  z,  t  in  the  equation  of 
motion 


-  dx*  -  dy*  -  dz* 


=  o 


of  a  point-mass  moving  uniformly  and  rectilinearly 
(i.e.  uninfluenced  by  gravity)  be  subjected  to  any 
transformation  corresponding  to  the  change  of  the 
x,  y,  z,  t  into  the  co-ordinates  xlt  x2>  xs,  -^  of  a  system 
of  reference  which  has  any  accelerated  motion 
whatsoever  with  regard  to  the  initial  system  x,  y, 
z,  t  ;  then,  in  general,  coefficients  g^v  will  occur 
in  the  transformed  expression  for  ds,  and  will  be 
functions  of  the  new  variables  xlf  .  .  .  #4,  so  that 
the  transformed  equation  will  be  : 


Taking  into  account  the  extended  region  of  validity 
of  this  equation,  one  will  be  able  to  regard  the 
g^  which  arise  from  the  accelerational  transforma- 
tion (vide  Note  25)  just  as  well,  as  due  to  the  action 
of  a  gravitational  field,  which  asserts  its  existence 
in  effecting  just  these  accelerations.  Gravitational 
problems  thus  resolve  into  the  general  science  of 
motion  of  a  relativity -theory  of  all  motions. 

By  thus  accentuating  the  equivalence  of  gravita- 


THEORY  OF  GRAVITATION  49 

tional  and  accelerational  events,  we  raise  the  funda- 
mental fact,  that  all  bodies  in  the  gravitational 
field  of  the  earth  fall  with  equal  acceleration,  to  a 
fundamental  assumption  for  a  new  theory  of  gravita- 
tional phenomena.  This  fact,  in  spite  of  its  being 
reckoned  amongst  the  most  certain  of  those  gathered 
from  experience,  has  hitherto  not  been  allotted  any 
position  whatsoever  in  the  foundations  of  mechanics. 
On  the  contrary,  the  Galilean  law  of  inertia  makes 
an  event  which  had  never  been  actually  observed 
(the  uniform  rectilinear  motion  of  a  body,  which  is 
not  subject  to  external  forces)  function  as  the 
main-pillar  amongst  the  fundamental  laws  of  me- 
chanics. This  brought  about  the  strange  view 
that  inertial  and  gravitational  phenomena,  which 
are  probably  not  less  intimately  connected  with 
one  another  than  electric  and  magnetic  phenomena, 
have  nothing  to  do  with  one  another.  The  phe- 
nomenon of  inertia  is  placed  at  the  base  of 
classical  mechanics  as  the  fundamental  property 
of  matter,  whereas  gravitation  is  only,  as  it  were, 
introduced  by  Newton's  law  as  one  of  the  many 
possible  forces  of  nature.  The  remarkable  fact  of  the 
equality  of  the  inertial  and  gravitational  mass  of 
bodies  only  appears  as  an  accidental  coincidence. 

Einstein's  principle  of  equivalence  assigns  to 
this  fact  the  rank  to  which  it  is  entitled  in  the 
theory  of  motional  phenomena.  The  new  equation 
of  motion  (i)  is  intended  to  describe  the  relative 
motions  of  bodies  with  respect  to  one  another 
under  the  influence  of  both  inertia  and  gravity. 
The  gravitational  and  inertial  phenomena  are  amal- 
gamated in  the  one  principle  that  the  motion  take 
place  in  the  geodetic  line  fflds  =  o).  Since  the 
element  of  arc 


ds 


=  J  S 
*  i 


50       THE  FOUNDATIONS  OF  EINSTEIN'S 

preserves  its  form  after  any  arbitrary  transformation 
of  the  variables,  all  systems  of  reference  are  equally 
justified  as  such,  i.e.  there  is  none  which  is  more 
important  than  any  other. 

The  most  important  part  of  the  problem,  with 
which  Einstein  saw  himself  confronted,  was  the 
setting-up  of  differential  equations  for  the  gravita- 
tional potentials  g^  of  the  new  theory.  With  the 
help  of  these  differential  equations,  the  gMl/s  were 
to  be  unambiguously  calculated  (i.e.  as  single- 
valued  functions)  from  the  distribution  of  the 
quantities  exciting  the  gravitational  field  ;  and  the 
motion  (e.g.  of  the  planets)  which  was  described, 
according  to  equation  (i)  by  inserting  these  values 
for  the  gnv's,  had  to  agree  with  the  observed  motion, 
if  the  theory  was  to  hold  true.  In  setting  up  these 
differential  equations  for  the  gravitational  potentials 
gMV  Einstein  made  use  of  hints  gathered  from 
Newton's  theory,  in  which  the  factor  which  excites 
the  field  in  Poisson's  equation  A</>  =  —  477/3  for  the 
Newtonian  gravitational  potential  (viz.  the  factor 
represented  by  /o,  the  density  of  mass  in  this  equa- 
tion) is  put  proportional  to  a  differential  expression 
of  the  second  order.  This  circumstance  prescribes, 
as  it  were,  the  method  of  building  up  these  equations, 
taking  for  granted  that  they  are  to  assume  a  form 
similar  to  that  of  Poisson's  equation. 

In  conformity  with  the  deepened  meaning  we 
have  assigned  to  the  mutual  relation  between 
inertia  and  gravity,  as  well  as  to  the  connection 
between  the  inertia  and  latent  energy  of  a  body, 
we  find  that  ten  components  of  the  quantity  which 
determines  the  "  energetic  "  state  at  any  point  of 
the  field,  and  which  was  already  introduced  by  the 
special  theory  of  relativity  as  "  stress-energy-tensor," 
duly  make  their  appearance  in  place  of  the  density 
of  mass  p,  in  Poisson's  equation. 


THEORY  OF  GRAVITATION  51 

Concerning  the  differential  expressions  of  the 
second  order  in  the  gM,/s  which  are  to  correspond 
to  the  A<£  of  Poisson's  equation,  Riemann  has  shown 
the  following  :  the  measure-relations  of  a  manifold 
based  on  the  line-element 

4 

ds2  =  Zg^dXpdx,,, 
i 

are  in  the  first  place  determined  by  a  differential 
expression  of  the  fourth  degree  (the  Riemann- 
Christoffel  Tensor),  which  is  independent  of  the 
arbitrary  choice  of  the  variables  xlf  .  .  .  x±  and 
from  which  all  other  differential  expressions  which 
are  likewise  independent  of  the  arbitrary  choice 
of  the  variables  xlt  .  .  .  #4  and  only  contain  the 
gpv's  and  their  derivatives,  can  be  developed  (by 
means  of  algebraical  and  differential  operations). 
This  differential  expression  leads  unambiguously, 
i.e.  in  only  one  possible  way,  to  ten  differential 
expressions  in  the  g^'s.  And  now,  in  order  to 
arrive  at  the  required  differential  equations,  Einstein 
puts  these  ten  differential  expressions  proportional 
to  the  ten  components  of  the  stress-energy-tensor, 
regarding  the  latter  ten  as  the  quantities  exciting 
the  field.  He  inserts  the  gravitational  constant 
as  the  constant  of  gravitation.  These  differential 
equations  for  the  gMV's,  together  with  the  principle 
of  motion  given  above,  represent  the  fundamental 
laws  of  the  new  theory.  To  the  first  order  they, 
in  point  of  fact,  lead  to  those  forms  of  motion, 
with  which  Newton's  theory  has  familiarized  us 
(vide  Note  26).  More  than  this,  without  requiring 
the  addition  of  any  further  hypothesis,  they  mathe- 
matically account  for  the  only  phenomenon  in  the 
theory  of  planetary  motion  which  could  not  be  ex- 
plained on  the  Newtonian  theory,  viz.  the  occur- 
rence of  the  remainder-term  in  the  expression 


52       THE  FOUNDATIONS  OF  EINSTEIN'S 

for  the  motion  of  Mercury's  perihelion.  Yet  we 
must  bear  in  mind  that  there  is  a  certain  arbitrariness 
in  these  hypotheses  just  as  in  that  made  for  the 
fundamental  law  of  motion.  Only  the  careful 
elaboration  of  the  new  theory  in  all  its  consequences, 
and  the  experimental  testing  of  it  will  decide  whether 
the  new  laws  have  received  their  final  forms. 

Since  the  formulae  of  the  new  theory  are  based 
upon  a  space-time-manifold,  the  line-element  of 
which  has  the  general  form 


*=,; 

all  other  physical  laws,  in  order  to  bring  the  general 
theory  of  relativity  to  its  logical  conclusion,  must 
receive  (see  p.  46)  a  form  which,  in  agreement 
with  the  new  measure-conditions,  must  be  indepen- 
dent of  the  arbitrary  choice  of  the  four  variables 

Mathematics  has  already  performed  the  pre- 
liminary work  for  the  solution  of  this  problem 
in  the  calculus  of  absolute  differentials  ;  Einstein 
has  elaborated  them  for  his  particular  purposes 
(in  his  essay  "  Concerning  the  formal  foundations 
of  the  general  theory  of  relativity  *  ")  ;  Gauss 
invented  the  calculus  of  absolute  differentials  in 
order  to  study  those  properties  of  a  surface  (in  the 
theory  of  surfaces)  which  are  not  affected  by  the 
position  of  the  surface  in  space  nor  by  inelastic 
continuous  deformations  of  the  surface  (deforma- 
tions without  tearing),  so  that  the  value  of  the 
line-element  does  not  alter  at  any  point  of  the 
surface.  As  such  properties  depend  upon  the  inner 
measure-relations  of  the  surface  only,  one  avoids 

*  "  Uber  die  formalen  Grundlagen  der  allgemeinen  Relativitats- 
theorie,"  Sit*.  Ber.  d.  Kgl.  Preuss.  Akad.  d.  Wiss.,  xli.,  1916,  S.  1080. 


THEORY  OF  GRAVITATION  53 

referring,  in  the  theory  of  surfaces,  to  the  usual 
system  of  co-ordinates,  i.e.  one  avoids  reference  to 
points  which  do  not  themselves  lie  on  the  surface. 
Instead  of  this,  every  point  in  the  surface  is  fixed, 
by  covering  the  surface  with  a  net-work,  consisting 
of  two  intersecting  arbitrary  systems  of  curves, 
in  which  each  curve  is  characterized  by  a  parameter  ; 
every  point  of  the  surface  is  then  unambiguously, 
i.e.  singly,  defined  by  the  two  parameters  of  the 
two  curves  (one  from  each  system)  which  pass 
through  it.  According  to  this  view  of  surfaces,  a 
cylindrical  envelope  and  a  plane,  for  instance, 
are  not  to  be  regarded  as  different  configurations  : 
for  each  can  be  unfolded  upon  the  other  without 
stretching,  and  accordingly  the  same  planimetry 
holds  for  both — a  criterion  that  the  inner  measure- 
relations  of  these  two  manifolds  are  the  same 
(vide  Note  27).  The  general  theory  of  relativity 
is  based  upon  the  same  view ;  but  now  not  as 
applied  to  the  two-dimensional  manifold  of  surfaces, 
but  with  respect  to  the  four-dimensional  space-time 
manifold.  As  the  four  space-time  variables  are 
devoid  of  all  physical  meaning,  and  are  only  to  be 
regarded  as  four  parameters,  it  will  be  natural 
to  choose  a  representation  of  the  physical  laws, 
which  provides  us  with  differential  laws  which 
are  independent  of  the  chance  choice  of  the  xlt 
%2>  xs>  %\  '>  this  is  what  is  done  by  the  calculus  of 
absolute  differentials.  The  results  of  the  preceding 
paragraphs,  the  far-reaching  consequences  of  which 
can  be  fully  recognized  only  by  a  detailed  study 
of  the  mathematical  developments  involved,  may 
be  summarized  as  follows  : 

A  mechanics  of  the  relative  motions  of  bodies, 
which  is  in  harmony  with  the  two  fundamental 
postulates  of  continuity  and  relativity,  can  be 
built  up  only  on  a  fundamental  law  of  motion 


54      THE  FOUNDATIONS  OF  EINSTEIN'S 

that  preserves  its  form  independently  of  the  kind 
of  motion  the  system  is  undergoing.  An  available 
law  of  this  kind  is  given  if  we  raise  the  law  of  motion 
along  a  geodetic  line,  which,  in  the  special  theory 
of  relativity,  holds  only  for  a  body  moving  under 
no  forces,  to  the  rank  of  a  general  differential  law 
of  the  motion  in  the  gravitational  field,  too.  In 
this  general  law,  we  must,  it  is  true,  give  the  line- 
element  of  the  orbit  of  the  moving  body  the  general 
form  : 

ds  = 


at  which  we  arrive  in  the  second  section,  using  as 
our  basis  the  two  fundamental  postulates.  The 
new  functions  g^  that  now  occur  may  be  -inter- 
preted as  the  potentials  of  the  gravitational  field, 
if  we  take  our  stand  on  the  hypothesis  of  equivalence. 
To  calculate  the  quantities  gMV  from  the  factors 
determining  the  gravitational  field,  namely,  matter 
and  energy,  it  immediately  suggests  itself  to  us 
to  assume  a  system  of  differential  equations  of  the 
second  order,  that  are  built  up  analogously  to 
Poisson's  differential  equation  for  the  Newtonian 
gravitational  potential.  These  differential  equa- 
tions, together  with  the  fundamental  law  of  motion, 
represent  the  fundamental  equations  of  the  new 
mechanics  and  the  theory  of  gravitation. 

Since  the  new  theory  uses  the  generalized  curvi- 
linear co-ordinates  xlt  x2)  x3,  #4,  and  not  the  Car- 
tesian co-ordinates  of  Euclidean  geometry,  all  the 
other  physical  laws  must  also  receive  a  general 
form  that  is  independent  of  the  special  choice  of 
co-ordinates.  The  mathematical  instrument  for  re- 
moulding these  formulae  is  given  by*the*general 
calculus  of  differentials. 

This  theory,   which  is  built  up  from  the  most 


THEORY  OF  GRAVITATION  55 

general  assumptions,  leads,  for  a  first  approxi- 
mation, to  Newton's  laws  of  motion.  Wherever 
deviations  from  the  theory  hitherto  accepted  reveal 
themselves,  we  have  possibilities  of  testing  the  new 
theory  experimentally.  Before  we  turn  to  this 
question,  let  us  look  back,  and  become  clear  as  to 
the  attitude  which  the  general  theory  of  relativity 
compels  us  to  adopt  towards  the  various  questions 
of  principle  we  have  touched  upon  in  the  course 
of  this  essay. 

(b)  RETROSPECT 

i.  The  conceptions  "  inertia! "  and  "  gravita- 
tional "  (heavy)  mass  no  longer  have  the  absolute 
meaning  which  was  assigned  to  them  in  Newton's 
mechanics.  The  "  mass  "  of  a  body  depends,  on 
the  contrary,  exclusively  upon  the  presence  and 
relative  position  of  the  remaining  bodies  in  the 
universe.  The  equality  of  inertial  and  gravita- 
tional mass  is  put  at  the  head  of  the  theory  as  a 
rigorously  valid  principle.  The  hypothesis  of  equi- 
valence herein  supplements  the  deduction  of  the 
special  theory  of  relativity,  that  all  energy  possesses 
inertia,  by  investing  all  energy  with  a  corresponding 
gravitation.  It  becomes  possible — on  the  basis 
(be  it  said)  of  certain  special  assumptions  into  which 
we  cannot  enter  here — to  regard  rotations  un- 
restrictedly as  relative  motions  too,  so  that  the 
centrifugal  field  around  a  rotating  body  can  be 
interpreted  as  a  gravitational  field,  produced  by 
the  revolution  of  all  the  masses  in  the  universe 
about  the  non-rotating  body  in  question.  In 
this  manner  mechanics  becomes  a  perfectly  general 
theory  of  relative  motions.  As,  our  statements 
are  concerned  only  with  observations  of  relative 
motions,  the  new  mechanics  fulfils  the  postulate 


56      THE  FOUNDATIONS  OF  EINSTEIN'S 

that  in  physical  laws  observable  things  only  are 
to  be  brought  into  causal  connection  with  one 
another.  It  also  fulfils  the  postulate  of  continuity  ; 
since  the  new  fundamental  laws  of  mechanics  are 
differential  laws,  which  contain  only  the  line- 
element  ds  and  no  finite  distances  between  bodies. 

2.  The  principle  of  the  constancy  of  the  velocity 
of  light  in  vacuo,  which  was  of  particular  importance 
in   the   special   theory   of   relativity,   is   no   longer 
valid  in  the  general  theory  of  relativity.     It  pre- 
serves its   validity   only  in   regions  in   which   the 
gravitational  potentials  are  constant,  finite  portions 
of  which  we  can  never  meet  with  in  reality.     The 
gravitational  field  upon  the  earth's  surface  is  cer- 
tainly so  far  constant  that  the  velocity  of  light, 
within  the  limits  of  accuracy  of  our  measurements, 
had  to  appear  to  be  a  universal  constant  in  the 
results  of  Michelson's  experiments.      In  a  gravita- 
tional field,   however,   in   which   the   gravitational 
potentials  vary  from  place  to  place,  the  velocity 
of  light  is  not  constant ;    the  geodetic  lines,  along 
which  light  propagates  itself,  will  thus  in  general 
be  curved.     The  proof  of  the  curvature  of  a  ray  of 
light,  which  passes  by  in  close  proximity  to  the  sun, 
offers  us  one  of  the  most  important  possibilities  of 
confirming  the  new  theory. 

3.  The  greatest  change  has  been  brought  about 
by  the  general  theory  of  relativity  in  our  concep- 
tions of  space  and  time.* 

According  to  Riemann  the  expression  for  the 
line-element,  viz. 

4 

1 

*  This  aspect  of  the  problem  has  been  treated  with  particular 
clearness  and  detail  in  the  book  "  Raum  und  Zeit  in  der  gegenwartigen 
Physik,"  by  Moritz  Schlick,  published  by  Jul.  Springer,  Berlin.  The 
Clarendon  Press  has  published  an  English  rendering  under  the  title : 
"  Space  and  Time  in  Contemporary  Physics." 


THEORY  OF  GRAVITATION  57 

determines,  in  our  case,  the  measure-relations  of 
the  continuous  space-time  manifold  ;  and  according 
to  Einstein  the  coefficients  g^  of  the  line-element 
ds  have,  in  the  general  theory  of  relativity,  the 
significance  of  gravitational  potentials.  Quantities, 
which  hitherto  had  only  a  purely  geometrical 
import,  for  the  first  time  became  animated  with 
physical  meaning.  It  seems  quite  natural  that 
gravitation  should  herein  play  the  fundamental 
part,  viz.  that  of  predominating  over  the  measure- 
laws  of  space  and  time.  For  there  is  no  physical 
event  in  which  it  does  not  co-operate,  inasmuch 
as  it  rules  wherever  matter  and  energy  come  into 
play.  Moreover,  it  is  the  only  force,  according  to 
our  present  knowledge,  which  expresses  itself  quite 
independently  of  the  physical  and  chemical  constitu- 
tion of  bodies.  It  therefore  without  doubt  occupies 
a  unique  position,  in  its  outstanding  importance 
for  the  construction  of  a  physical  picture  of  the 
world. 

According  to  Einstein's  theory,  then,  gravitation 
is  the  "  inner  ground  of  the  metric  relations  of  space 
and  time  "  in  Riemann's  sense  (vide  the  final  para- 
graph of  Riemann's  essay  "  On  the  hypotheses 
which  lie  at  the  bases  of  geometry "  quoted  on 
p.  29).  If  we  uphold  the  view  that  the  space- 
time  manifold  is  continuously  connected,  its  measure- 
relations  are  not  then  already  contained  in  its 
definition  as  being  a  continuous  manifold  of  the 
dimensions  "  four."  These  have,  on  the  contrary, 
yet  to  be  gathered  from  experience.  And  it  is, 
according  to  Riemann,  the  task  of  the  physicist 
finally  to  seek  the  inner  ground  of  these  measure- 
relations  in  "  binding  forces  which  act  upon  it." 
Einstein  has  discovered  in  his  theory  of  gravitation 
a  solution  to  this  problem,  which  was  presumably 
first  put  forward  in  such  clear  terms  by  Riemann. 


58      THE  FOUNDATIONS  OF  EINSTEIN'S 

At  the  same  time  he  gives  an  answer  to  the  question 
of  the  true  geometry  of  physical  space,  a  question 
which  has  exercised  physicists  for  the  last  century, 
— but  an  answer,  it  is  true,  of  a  sort  quite  different 
from  that  which  had  been  expected. 

The  alternative,  Euclidean  or  non-Euclidean 
geometry,  is  not  decided  in  favour  of  either  one  or 
the  other ;  but  rather  space,  as  a  physical  thing 
with  given  geometrical  properties,  is  banished  out 
of  physical  laws  altogether :  just  as  ether  was 
eliminated  out  of  the  laws  of  electrodynamics  by 
the  Lorentz-Einstein  special  theory  of  relativity. 
This,  too,  is  a  further  step  in  the  sense  of  the 
postulate  that  only  observable  things  are  to  have 
a  place  in  physical  laws.  The  inner  ground  of 
metric  relations  of  the  space- time  manifold,  in 
which  all  physical  events  take  place,  lies,  according 
to  Einstein's  view,  in  the  gravitational  conditions. 
Owing  to  the  continual  motion  of  bodies  relatively 
to  one  another,  these  gravitational  conditions  are 
continually  altering ;  and,  therefore,  one  cannot 
speak  of  an  invariable  given  geometry  of  measure 
or  distance — whether  Euclidean  or  non-Euclidean. 
As  the  laws  of  physics  preserve  their  form  in  the 
general  theory  of  relativity,  independent  of  how 
the  four  variables  xlt  .  .  .  #4  may  chance  to  be 
chosen,  the  latter  have  no  absolute  physical  meaning. 
Accordingly  xlt  xz,  x3,  for  instance,  will  not  in 
general  denote  three  distances  in  space  which  can 
be  measured  with  a  metre  rule,  nor  will  %±  denote 
a  moment  of  time  which  can  be  ascertained  by 
means  of  a  clock.  The  four  variables  have  only 
the  character  of  numbers,  parameters,  and  do 
not  immediately  allow  of  an  objective  interpretation. 
Time  and  space  have,  therefore,  not  the  meaning 
of  real  physical  things  in  the  description  of  the 
events  of  physical  nature. 


THEORY  OF  GRAVITATION  59 

And  yet  it  seems  as  if  the  new  theory  may  even 
be  able  to  give  a  definite  answer  in  favour  of  one 
or  other  of  the  above  alternatives,  if,  namely,  we 
postulate  their  validity  for  the  world  as  a  whole. 
The  application  of  the  formulae  of  the  new  theory 
to  the  world  as  a  whole  at  first  led  to  the  same 
difficulties  as  those  revealed  in  classical  mechanics. 
Boundary  conditions  for  what  is  infinitely  distant 
could  not  be  set  up  entirely  satisfactorily  and  at 
the  same  time  satisfy  the  condition  of  general 
relativity.  Yet  Einstein  *  succeeded  in  extending 
the  differential  equations  for  the  gravitational 
potentials  g^  in  such  a  way  that  it  became  possible 
to  apply  his  theory  of  gravitation  to  the  universe. 
The  difficulties  that  arose  for  the  boundary  conditions 
at  infinity  here  vanished,  for  an  extraordinarily 
interesting  reason.  For  it  was  shown  that  in 
these  new  formulae  a  space  that  is  filled  uniformly 
with  matter  which  is  at  rest  would,  to  a  first  approxi- 
mation, be  built  up  like  an,  indeed,  unbounded,  but 
finitely  closed  space,  so  that  boundary  conditions 
would  not  appear  at  all  for  infinity.  Even  if  the 
assumptions  that  would  lead  to  this  result  are  not 
fulfilled  in  the  world,  yet  it  must  be  remembered 
that  the  velocities  of  matter  as  ascertained  in  the 
case  of  the  stars  are  extraordinarily  small  com- 
pared with  the  velocity  of  light  which  we  now 
take  as  our  unit.  Nor  does  the  distribution  of  the 
matter  so  far  show,  in  the  main,  irregularities 
sufficient  to  place  Einstein's  view  of  a  stationary, 
uniformly-filled  world  quite  out  of  the  realm  of 
possible  truth.  Thus  this  deduction  of  the  theory 
would  answer  our  above  alternative  in  this  sense  : 
the  geometry  that  we  must  use  as  our  basis  of 
spatial  happening  is,  indeed,  neither  Euclidean  nor 

*  "  Kosmologische  Betrachtungen  zur    allgemeinen    Relativitats- 
theorie  "     Sitz.  Ber.  d.  Preuss.  Akad.  der  Wiss.,  1917,  p.  142. 


60       THE  FOUNDATIONS  OF  EINSTEIN'S 

non-Euclidean,  but,  as  stated  above,  conditioned 
by  the  gravitational  states  from  place  to  place. 
But  a  world  built  up  according  to  the  simplest 
scheme  would  in  the  new  theory  behave  on  the 
whole  like  a  finite  closed  manifold,  that  is,  as  if  it 
were  non-Euclidean.  Even  if  this  result  is  only 
of  theoretical  importance  for  the  present,  since  the 
stellar  system  that  we  see  around  us  does  not  fulfil 
Einstein's  assumptions — in  particular,  the  scarcely- 
to-be-doubted  flattening  of  the  Milky  Way  is  not 
compatible  with  these  simple  assumptions — and 
since  we  have  at  present  no  knowledge  of  the 
stellar  systems  outside  the  Milky  Way,  yet  this 
aspect  of  the  theory  opens  up  undreamed-of  per- 
spectives for  our  view  of  the  world  as  a  whole. 

4.  The  gravitational  theory,  which  emerges  out 
of  the  general  theory  of  relativity,  is,  in  contra- 
distinction to  the  Newtonian  theory,  built  up,  not 
upon  an  elementary  law  of  the  gravitational  forces, 
but  upon  an  elementary  law  of  the  motion  of  a  body 
in  the  gravitational  field.  Consequently,  the  ex- 
pressions which  would  be  interpreted  as  gravitational 
forces  in  the  new  theory  play  only  a  minor  part  in 
the  building-up  of  the  theory  (as  indeed  the  con- 
ception of  force  in  mechanics  altogether  is  to  be 
regarded  as  only  an  auxiliary  or  derived  conception, 
if  we  regard  it  as  the  object  of  mechanics  to  give 
a  flawless  description  of  the  motions  occurring  in 
physical  events). 

Nor  does  Einstein's  theory  endeavour  to  explain 
the  nature  of  gravitation  ;  it  does  not  seek  to  give 
a  mechanical  model,  which  would  symbolize  the 
gravitational  effect  of  two  masses  upon  one  another. 
This  is  what  the  various  theories  involving  ether- 
impulses  attempted  to  do,  by  freely  using  hypothet- 
ical quantities  which  had  never  been  actually 
observed,  such  as  ether-atoms.  It  is  very  doubtful 
whether  such  endeavours  will  ever  lead  to  a  satis- 


THEORY  OF  GRAVITATION  61 

factory  theory  of  gravitation.  For,  the  difficulties 
of  Newton's  mechanics  are  not  contained  only  in 
the  fact  that  it  formulates  the  law  of  gravitation 
as  a  law  of  forces  acting  at  a  distance.  Two  much 
more  serious  points  are  :  first,  that  the  close  rela- 
tionship existing  between  inertial  and  gravitational 
phenomena  receives  no  recognition  whatsoever, 
although  Newton  was  already  aware  of  the  fact 
that  inertial  and  gravitational  mass  are  equal ; 
and  second,  that  Newton's  mechanics  does  not 
present  us  with  a  theory  of  the  relative  motions 
of  bodies,  although  we  only  observe  relative  motions 
of  bodies  with  respect  to  one  another.  Re-moulding 
Newton's  law  of  gravitational  force,  in  order  to 
make  the  attraction  of  matter  more  feasible,  would 
therefore  not  have  helped  us  finally  to  a  satisfactory 
theory  of  the  phenomena  of  motion  (vide  Note  28). 

What  distinguishes  the  Newtonian  theory,  above 
all,  is  the  extraordinary  simplicity  of  its  mathe- 
matical form.  Classical  mechanics,  which  is  built 
up  on  Newton's  initial  construction,  will,  for  this 
reason,  never  lose  its  importance  as  an  excellent 
mathematical  theory  for  arithmetically  following 
the  observed  phenomena  of  motion. 

Einstein's  theory,  on  the  other  hand,  as  far  as 
the  uniformity  of  its  conceptual  foundations  is 
concerned,  satisfies  all  the  conditions  for  a  physical 
theory.  The  fact  that  (by  abandoning  the  Euclidean 
measure  of  distance)  it  cuts  its  connection  with  the 
familiar  representation  by  means  of  Cartesian  co- 
ordinates, will  not  be  felt  to  be  a  disturbing  factor, 
as  soon  as  the  analytical  appliances,  which  have 
been  called  into  use  as  a  help,  have  been  more 
generally  adopted.  This  mathematical  elaboration 
of  the  theory  at  the  same  time  gives  to  the  astro- 
nomer the  task  of  testing  the  theory  experimentally 
in  those  phenomena  in  which  measurable  deviations 
from  the  results  of  the  classical  theory  arise. 


62       THE  FOUNDATIONS  OF  EINSTEIN'S 


§6 

THE  VERIFICATION  OF  THE  NEW  THEORY  BY 
ACTUAL  EXPERIENCE 

AS  far  as  can  be  seen  at  present,  there  are 
three  possible  experiments  for  verifying  Eins- 
tein's theory  of  gravitation  ;  all  three  can 
be  performed  only  by  the  agency  of  astronomy. 
One  of  them — arising  from  the  deviation  of  the 
motion  of  a  material  point  in  the  gravitational 
field  according  to  Einstein's  theory,  as  compared 
with  that  required  by  Newton's  theory — has  already 
decided  in  favour  of  the  new  theory  :  not  less  so 
one  of  the  other  two  that  arise  through  a  com- 
bination of  electromagnetic  and  gravitational  phen- 
omena. 

Since  the  sun  far  exceeds  all  other  bodies  of  the 
solar  system  in  mass,  the  motion  of  each  particular 
planet  is  primarily  conditioned  by  the  gravitational 
field  of  the  sun.  Under  its  action  the  planet  de- 
scribes, according  to  Newton's  theory,  an  ellipse 
(Kepler's  law),  the  major  axis  of  which — defined 
by  connecting  the  point  of  its  path  nearest  the  sun 
(perihelion)  with  the  farthest  point  (aphelion) — is 
at  rest,  relative  to  the  stellar  system.  Upon  this 
elliptic  motion  of  a  planet  there  are  superimposed 
more  or  less  considerable  influences  (disturbances) 
due  to  the  remaining  planets,  which  do  not,  how- 
ever, appreciably  alter  the  elliptic  form ;  these 
influences  partly  only  call  forth  periodical  fluctua- 


THEORY  OF  GRAVITATION  63 

tions  in  the  defining  elements  of  the  initial  ellipse 
(i.e.  major  axis,  eccentricity,  etc.),  partly  cause  a 
continual  increase  or  decrease  of  the  latter.  In 
this  second  kind  of  disturbance  are  also  to  be  classed 
the  slow  rotation  of  the  major  axis,  and  consequently 
also  of  the  corresponding  perihelion,  relative  to 
the  stellar  system  ;  which  has  been  observed  in 
the  case  of  all  planets.  For  all  the  larger  planets, 
the  observed  motions  of  the  perihelion  agree  with 
those  calculated  from  the  disturbing  effects  (except 
for  small  deviations  which  have  not  been  definitely 
established,  as  in  the  case  of  Mars)  ;  on  the  other 
hand,  in  the  case  of  Mercury  the  calculations  give 
a  value  which  is  too  small  by  43"  per  100  years. 
Hypotheses  of  the  most  diverse  description  have 
been  evolved  to  explain  this  difference  ;  but  all 
of  them  are  unsatisfactory.  They  oblige  one  to 
resort  to  still  unknown  masses  in  the  solar  system  : 
and,  as  all  the  searches  for  masses  large  enough  to 
explain  this  anomalous  behaviour  of  Mercury  prove 
fruitless,  one  is  compelled  to  make  assumptions 
about  the  distribution  of  these  hypothetical  masses, 
in  order  to  excuse  their  invisibility.  In  view  of 
these  circumstances,  there  is  no  shade  of  probability 
in  these  hypotheses. 

According  to  Einstein's  theory,  a  planet,  at  the 
distance  of  Mercury  for  instance,  moves,  under 
the  action  of  the  sun's  attraction,  along  the  "  straight- 
est  path,"  according  to  the  equation  : 


The  g^'s  can  be  derived  from  the  differential 
equations,  which  were  given  for  them  above,  and 
which  result  from  the  assumed  sole  presence  of  the 
sun  and  the  planet  being  regarded  as  a  mass  concen- 
trated at  a  point.  Einstein's  developments  give  the 


64      THE  FOUNDATIONS  OF  EINSTEIN'S 

ellipse  of  Kepler  too  as  a  first  approximation  for 
the  path  of  the  planet :  at  a  higher  degree  of  ap- 
proximation, however,  it  is  found  that  the  radius 
vector  from  the  sun  to  the  planet,  between  two 
consecutive  passages  through  perihelion  and  aphe- 
lion, sweeps  out  an  angle,  which  is  about  0-05" 
greater  than  180° ;  so  that,  for  each  complete 
revolution  of  the  planet  in  its  path,  the  major  axis 
of  the  path — i.e.  the  straight  line  connecting  perihe- 
lion with  aphelion — turns  through  about  O'i"  in 
the  sense  in  which  the  path  is  described.  There- 
fore, in  100  years — Mercury  completes  a  revolution 
in  88  days — the  major  axis  will  have  turned  through 
43".  The  new  theory,  therefore,  actually  explains 
the  hitherto  inexplicable  amount,  43  seconds  per 
100  years,  in  the  motion  of  Mercury's  perihelion, 
merely  from  the  effect  of  the  sun's  gravitation. 
(The  deviations  due  to  such  disturbances  would 
only  differ  very  inappreciably  from  the  values 
obtained  by  Newton's  theory  in  the  case  of  the 
remaining  planets.)  The  only  arbitrary  constant 
which  enters  into  these  calculations  is  the  gravita- 
tional constant  which  figures  in  the  differential 
equations  for  the  gravitational  potentials  g^,  as  has 
already  been  mentioned  on  page  50.  This  achieve- 
ment of  the  new  theory  can  scarcely  be  estimated 
too  highly. 

The  reason  that  a  measurable  deviation  from  the 
results  according  to  Newton's  theory  occurs  in  the 
case  of  Mercury,  the  planet  nearest  to  the  sun, 
but  not  in  the  case  of  the  planets  more  distant 
from  the  sun,  is  that  this  deviation  decreases  rapidly 
with  increasing  distance  of  the  planet  from  the  sun, 
so  that  it  already  becomes  imperceptible  at  the 
distance  of  the  earth.  In  the  case  of  Venus,  the 
eccentricity  of  the  path  is,  unfortunately,  so  small, 
that  it  scarcely  differs  from  a  circle :  and  the 


THEORY  OF  GRAVITATION  65 

position  of  the  perihelion  can,  therefore,  only  be 
determined  with  great  uncertainty. 

Of  the  other  two  possibilities  of  verifying  the 
theory,  one  arises  from  the  influence  of  gravitation 
upon  the  time  an  event  takes  to  pass.  How  such 
an  influence  can  come  about,  will  be  evident  from 
the  following  example :  According  to  the  new 
theory,  an  observer  cannot  immediately  distinguish 
whether  a  change,  which  he  observes  during  the 
passage  of  a  certain  event,  is  due  to  a  gravitational 
field  or  to  a  corresponding  acceleration  of  his  place 
of  observation  (his  system  of  reference).  Let  us 
assume  an  invariable  gravitational  field,  denoted 
by  parallel  lines  of  force  in  the  negative  direction 
of  the  2-axis,  and  having  a  constant  value  y  for  the 
acceleration  with  which  all  bodies  in  the  field  fall 
(i.e.  characterized  by  conditions  which  approxi- 
mately exist  on  the  surface  of  the  earth).  According 
to  Einstein's  theory,  any  event  will  take  place  in 
this  field  in  just  the  same  way  as  it  appears  to  occur 
when  referred  to  a  co-ordinate  system  which  has 
an  acceleration  y  in  the  positive  direction  of  the 
<z-axis.  Now  if  a  ray  of  light,  the  time  of  oscillation 
of  which  is  vlf  travels  from  a  point  A — which  is  to  be 
conveniently  supposed  at  rest  relatively  to  the  cor- 
responding co-ordinate  system  at  the  moment  of 
departure  of  the  ray — in  the  direction  of  the  z-axis 
for  a  distance  h  to  a  point  B,  then  an  observer 
at  B  will,  owing  to  his  own  acceleration,  y,  have 
attained  a  velocity  y  .  h/c  at  the  instant  the  ray 
of  light  reaches  him  (c  denotes  the  velocity  of  light). 
According  to  the  usual  Doppler  Principle,  he  will 
assign  a  time  of  oscillation  v2  =  ^  (i  -f  y  .  h/c2) 
to  the  ray  of  light  as  a  first  approximation,  instead 
of  i/j.  If  we  transfer  the  same  event  to  the  equivalent 
gravitational  field,  this  result  assumes  the  following 
form :  The  time  of  oscillation  v2  of  a  ray  of  light 
5 


66        THE  FOUNDATIONS  OF  EINSTEIN'S 

at  a  place  B,  the  gravitational  potential  of  which 
differs  from  that  of  a  place  A  by  the  amount  +  <£, 
is  connected  with  the  time  of  oscillation  there 
observed  by  the  relation  : 


according  to  the  principle  of  equivalence  of  Einstein's 
theory  of  gravitation. 

This  special  case  shows  how  the  duration  of  an 
event  is  to  be  understood  as  being  dependent  upon 
the  gravitational  condition. 

Moreover,  one  can  regard  every  vibrating  system 
(which  emits  a  spectral  line)  as  a  clock,  the  motion 
of  which,  according  to  the  investigation  made  just 
above,  depends  upon  the  gravitational  potentials 
of  the  place  where  it  is  stationed.  This  same 
"  clock  "  will  have  a  different  time  of  oscillation 
at  another  place  in  the  field  according  to  the  gravita- 
tional potential,  i.e.  it  will  go  at  a  different  rate. 
Consequently,  a  particular  line  in  the  spectrum  of 
the  light  which  comes  from  the  sun,  e.g.  an  Fe-line 
(iron),  must  appear  to  be  shifted  in  comparison 
with  the  corresponding  line  as  produced  by  a 
source  of  light  (arc-lamp)  on  the  earth  ;  the  gravita- 
tional potential  at  the  surface  of  the  sun  has,  cor- 
responding to  the  latter's  great  mass,  a  different 
value  from  that  at  the  surface  of  the  earth,  and  a 
definite  time  of  oscillation  (colour)  is  characterized 
in  the  spectrum  by  a  definite  position  (Fraunhofer 
line).  It  has  not  yet  been  possible  to  observe 
this  effect,  which  amounts  to  about  o-oo8A  *  for 
a  wave-length  of  400^  with  certainty.  For  the 
conditions  of  emission  of  the  light  from  the  sun's 
surface  have  not  yet  been  sufficiently  investigated, 
and  the  systematic  errors  in  the  wave-lengths  in 

*  A  =  Angstrom  unit  =  jo  ~  8  cm. 


THEORY  OF  GRAVITATION  67 

the  light  from  the  source  used  for  comparison  on 
the  earth,  the  arc-lamp,  are  not  yet  sufficiently 
known  to  allow  the  negative  results  of  observation 
hitherto  obtained  to  be  regarded  as  giving  binding 
decisions.  This  is  the  more  true  inasmuch  as  in 
the  case  of  the  fixed  stars  there  are,  doubtless, 
signs  of  the  presence  of  a  gravitational  shift  of  the 
spectral  lines  (vide  the  closing  essay  of  this  book). 
It  is  a  particularly  important  task  of  astronomy  to 
establish  this  effect  with  certainty,  for  this  gravita- 
tional displacement  of  the  spectral  lines  is  a  direct 
consequence  of  the  hypothesis  of  equivalence,  and 
does  not  assume  the  other  hypotheses  of  the  theory 
such  as,  for  example,  the  differential  equations  of 
the  gravitational  field. 

The  third  and  particularly  important  inference 
from  Einstein's  theory  is  the  dependence  of  the 
velocity  of  light  upon  the  gravitational  potential, 
and  the  resultant  curvature  (based  upon  Huygens' 
principle)  of  a  ray  of  light  in  passing  through  a 
gravitational  field.  The  theory  asserts  that  a  ray 
of  light,  coming  e.g.  from  a  fixed  star,  and  which 
passes  in  close  proximity  to  the  sun,  has  a  curved 
path.  As  a  consequence  of  this  curvature,  the  star 
must  appear  displaced  from  its  true  position  in 
the  heavens  by  an  amount  which  attains  the  value 
17"  at  the  edge  of  the  sun's  disc,  and  decreases 
in  proportion  to  the  distance  from  the  centre  of 
the  sun.  But  since  a  ray  of  light  which  comes 
from  a  fixed  star  and  passes  by  the  sun  can  be 
caught  only  when  the  light  of  the  sun,  which  over- 
powers all  else  by  its  brilliancy,  is  intercepted 
before  its  entrance  into  our  atmosphere,  only  the 
rare  moments  of  a  total  eclipse  come  into  account 
for  this  observation  and  for  the  solution  of  the 
problem.  The  solar  eclipse  of  2Qth  May,  1919, 
during  which  photographs  were  taken  at  two  widely- 


68      THE  FOUNDATIONS  OF  EINSTEIN'S 

separated  stations,  for  the  purpose  of  this  test,  has, 
as  far  as  the  results  of  measurement  allow  us  to 
pass  definite  judgment,  decided  in  favour  of  the 
general  theory  of  relativity.* 

The  experimental  verification  of  Einstein's  theory 
of  gravitation  has  thus  not  reached  completion. 
But  if,  in  spite  of  this,  the  theory  can,  even  at  this 
early  stage,  justly  claim  general  attention,  the 
reason  is  to  be  found  in  the  unusual  unity  and  logical 
structure  of  the  ideas  underlying  it.  In  truth,  it 
solves,  at  one  stroke,  all  the  riddles,  concerning  the 
motions  of  bodies,  which  have  presented  them- 
s*elves  since  the  time  of  Newton,  as  the  result  of  the 
conventional  view  about  the  meaning  of  space  and 
time  in  the  physical  description  of  natural  pheno- 
mena. 

*  The  results  were  made  public  at  the  meeting  of  the  Royal 
Society  on  the  6th  Nov.,  1919.— H.  L.  B. 


THEORY  OF  GRAVITATION  69 


APPENDIX 

Note  i  (p.  4).  So  long  as  the  universal  signi- 
ficance of  the  velocity  of  light  remained  unknown, 
two  conjectures  were  possible  in  the  question  as 
to  whether,  under  certain  circumstances,  the  motion 
of  the  source  of  light  would  make  itself  observable 
in  the  velocity  of  propagation  of  light.  It  might 
be  surmised  that  the  velocity  of  the  source  simply 
added  itself  to  that  velocity  of  light  which  is  char- 
acteristic for  the  propagation  of  the  light  from  a  source 
at  rest.  Or,  it  might  be  conceived  that  the  motion 
of  the  source  has  no  influence  at  all  on  the  velocity 
of  the  light  emitted  by  it.  In  the  second  case  it 
was  imagined  that  the  source  of  light  only  excites 
the  periodically  changing  states  of  the  luminiferous 
ether,  which  is  at  rest,  that  is,  which  does  not 
share  in  the  motion  of  the  matter  (source  of  light), 
and  ^  that  these  states  then  propagate  themselves 
with  a  velocity  that  is  characteristic  of  the  ether,  and 
with  a  velocity  that  makes  these  states  perceptible  to 
us  as  light  waves.  This  view  had  finally  apparently 
won  the  day.  It  was  the  advent  of  the  special 
theory  of  relativity  and  the  quantum  hypothesis 
that  made  this  view  impossible.  For  the  special 
theory  of  relativity,  in  robbing  the  assertion : 
"  the  ether  is  at  rest  "  of  its  significance,  since  we 
may  arbitrarily  define  any  system  as  being  at  rest  in 
the  ether,  as  far  as  uniform  translations  are  concerned, 
and  in  depriving  the  luminiferous  ether  of  its  exis- 
tence, deprived  light-waves  of  their  carrying  or 


70      THE  FOUNDATIONS  OF  EINSTEIN'S 

transmitting  medium.  The  quantum  hypothesis,  in 
raising  light-quanta  to  the  rank  of  self-supporting 
individuals,  deprived  the  velocity  of  light  of  its 
character  as  a  constant  that  is  characteristic  of  the 
ether.  Thus,  our  view  of  light-quanta  again  leads 
to  a  kind  of  emission  theory  of  light.  According 
to  classical  mechanics  it  would  have  been  typical 
of  a  theory  of  emission  for  the  velocity  of  the  source 
in  motion  to  have  added  itself  to  the  velocity  of  the 
light  from  the  source  at  rest.  We  thus  revert 
to  the  conjecture  which  we  quoted  first  above. 
Now,  such  a  superposition  of  velocities  would 
necessarily  cause  quite  remarkable  phenomena  in 
the  case  of  spectroscopic  binary  stars  (de  Sitter, 
"  Phys.  Zeitschrift,"  14,  429).  For  if  two  stars  move 
in  circular  Kepler  orbits  around  each  other,  and  if 
our  line  of  sight  lies  in  the  common  plane  of  the 
orbits,  then  we  should  necessarily  perceive  the 
following  :  if  2T  is  the  time  of  revolution  of  the 
system,  v  the  orbital  velocity  of  the  one  (bright) 
component,  A  the  distance  of  the  whole  system 
from  the  earth,  and,  finally,  c,  the  velocity  in  vacuo 
of  the  light  from  the  source  which  is  at  rest,  then 
the  velocity  of  light  at  the  epoch  of  greatest  positive 
velocity  in  the  direction  of  vision  is  c  -\-  v,  and 
c  —  v  in  the  other  direction,  respectively.  Con- 
sequently the  time-interval  between  two  such  suc- 

cessive positions  would  have   the  values  T  -f  —  — 


and  T  —      —  alternately,  for  the  observer  on  the 
c2 

earth,   as   a   simple   calculation  shows.     Since,    on 
account    of    the    gigantic    distances    between    the 


fixed   stars,    the   member       -   may    become    very 

c 

great,  indeed,  greater  than  T,  we  should  be  able  to 
observe  definite  anomalies  in  the  case  of  the  spectro- 


THEORY  OF  GRAVITATION  71 

scopic  binaries.  For  the  time  intervals  between 
two  such  successive  epochs  in  the  orbit  should  be 
able  to  contract  to  nil,  indeed,  even  become  negative, 
and  we  should  not  be  able  to  interpret  the  measured 
Doppler  effects  by  means  of  motions  in  the  Kepler 
ellipses.  In  reality,  however,  these  anomalies  have 
never  manifested  themselves.  Observation  of  these 
very  sensitive  subjects  of  test  (spectroscopic  binaries) 
teaches  us  that  the  motion  of  the  source  of  light 
does  not  make  itself  remarked  in  the  propagation 
of  the  light.  This  renders  our  first  view  likewise 
untenable.  The  special  principle  of  relativity,  alone 
in  postulating  the  constancy  of  the  velocity  of  light, 
and  in  putting  forward  a  new  addition  theorem 
of  velocities,  has  led  us  to  an  attitude  in  this 
question  that  is  free  from  inner  contradictions 
and  compatible  with  experience.  „  (Cf.  Note  2.) 

Note  2  (p.  5).  There  are  essentially  two  funda- 
mental optical  experiments  on  which  our  view  of 
the  distinctive  significance  of  the  velocity  of  light 
in  physical  nature  is  founded  :  Fizeau's  experiment 
concerning  the  velocity  of  light  in  flowing  water, 
and  the  Michelson-Morley  experiment.  Aberration, 
on  the  other  hand,  has  nothing  to  do  directly  with 
the  question  whether  it  is  possible  to  prove  by 
means  of  optical  experiments  in  the  laboratory  a 
motion  of  the  earth  relative  to  the  ether.  The 
aberration  in  the  case  of  stars  states  merely  that 
the  motion  of  the  earth  relatively  to  the  star  under 
consideration  changes  periodically  in  the  course  of 
a  year.  If,  however,  we  hold  the  view  that  an 
all-pervading  ether  is  the  carrier  for  the  propagation 
of  the  light,  the  phenomenon  of  aberration  may 
be  satisfactorily  explained  only  if  we  assume  that 
this  ether  does  not  participate  in  the  motion  of  the 
earth. 

Fizeau's    experiment    was    designed    to    decide 


n      THE  FOUNDATIONS  OF  EINSTEIN'S 

finally  whether  moving  matter  influences  the  ether 
and  to  determine  the  value  of  the  velocity  of  light 
in  moving  matter  with  respect  to  the  observer. 
Michelson  and  Morley  repeated  the  experiment  in 
the  following  improved  form.  A  beam  of  light 
from  a  source  on  the  earth  is  sent  through  a  U-shaped 
tube,  through  which  water  flows,  in  the  direction 
of  both  limbs.  After  each  part  of  the  beam  has 
traversed  the  flowing  water,  the  one  in  the  direction  of 
the  current,  the  other  contrary  to  it,  the  two  beams, 
are  made  to  interfere.  The  light  and  the  water  move 
in  the  same  direction  in  the  one  limb,  and  oppo- 


FlG.   I. 

sitely  in  the  other.  Now,  there  immediately  ap- 
pear to  be  two  possibilities.  Either  the  water  that 
flows  with  a  velocity  v  with  respect  to  the  walls 
of  the  tube  drags  along  the  carrier  that  effects 
the  transmission  of  the  light,  namely  the  ether ; 

in  this  case,  the  velocity  of  the  light  is  -  -+-  v  in  the 

n 

one  limb,  and  —  —  v  in  the  other,  for,  on  account 
n 

of  the  coefficient  of  refraction  n  of  the  water,  -  is 

n 

the  velocity  of  light  in  resting  water.  Or,  the 
motion  of  the  water  has  no  influence  at  all  on  the 


THEORY  OF  GRAVITATION  73 

ether  which  transmits  light  and  which  permeates 

the  water.     In  this  case  the  velocity  of  light  is  - 

n 

in  both  limbs.  According,  as  the  one  or  the  other 
of  these  two  assumptions  is  valid,  the  interference 
fringes  would  have  to  become  displaced  or  remain 
at  rest  when  the  direction  of  the  current  is  reversed. 
The  experiment  decided  in  favour  of  neither  of  these 
possibilities.  The  interference  fringes  did,  indeed, 
become  shifted,  not  to  the  expected  amount,  how- 
ever, but  only  to  an  amount  that  would  result  if 

the  ether  assumes  the  velocity  v(i  —  -ij  in  water, 

and  not  the  full  value  v.  This  value  of  the  con- 
vection of  the  ether  is  called  Fresnel's  convection 
coefficient.  Yet  this  term  is  capable  of  being  mis- 
understood inasmuch  as  in  the  electrodynamics 
developed  by  Lorentz,  the  result  of  Fizeau's  experi- 
ment speaks  in  favour  of  an  ether  that  is  absolutely 
at  rest,  and  the  so-called  convection  coefficient  is 
only  a  consequence  of  the  structure  of  matter,  in 
particular  of  the  interaction  between  electrons  and 
matter,  a  question  into  which  we  cannot  enter  here. 
At  any  rate,  at  the  time  preceding  the  Michelson- 
Morley  experiment  aberration,  as  well  as  Fizeau's 
experiment,  appeared  to  speak  in  favour  of  an  ether 
that  was  absolutely  at  rest. 

Now,  the  Michelson-Morley  experiment  was  to 
establish  the  existence  of  the  current  of  ether 
(ether  "  wind  ")  through  which  the  earth  continually 
moves,  since  the  ether  is  supposed  not  to  participate 
in  the  motion  of  the  earth.  The  scheme  of  the 
experiment  is  as  on  p.  74. 

A  ray  of  light,  starting  out  from  L,  traverses  the 
course  LP  +  PSX  +  SXP  +  PFl :  here  Sj  and  S2  are 
two  mirrors,  on  to  which  the  ray  falls  perpendicu- 
larly ;  P  is  a  glass  plate  that  reflects  one  half  of 


74      THE  FOUNDATIONS  OF  EINSTEIN'S 

the  light  and  allows  the  remainder  to  pass  through  ; 
F  is  the  telescope  of  the  observer.  Another  ray  of 
light  traverses  the  course  LP  +  PS2  +  S2  P  +  PF. 
Let  PSi  =  PS2  =  /.  Further,  let  FSi  be  in  the 
direction  of  the  earth's  motion.  Our  assumption  is 
that  the  ether  does  not  share  in  the  earth's  motion. 
Let  the  velocity  of  the  earth  be  q.  Then  the 


FIG.  2. 


velocity   of   the   light   relative   to   the  instrument 
(earth)  is  as  follows  in  the  directions  specified  : 


PSt  :  c  -  q,  thus  the  light- time  for  the  journey  is 


PS2  : 
S2P  : 


Consequently,  the  course  PS±  +  SXP  is  traversed  in 
the  time 


THEORY  OF  GRAVITATION  75 

and  the  course  PS2  +  S2P  in  the  time 

t          2l       _ 
^-x/crzr^-  c 

The  difference  of  these  two  times  is 


If  we  exchange  the  positions  of  Sx  and  S2  by  turning 
the  whole  apparatus  through  90°,  then 


If  we  make  these  two  rays  of  light  interfere  at 
F  then,  when  the  apparatus  is  turned  through  90°, 
the  interference  fringes  should  become  shifted. 
The  amount  of  this  displacement  may  easily  be 
calculated.  If  we  denote  by  r  the  vibration  fre- 
quency of  the  light-ray  used  in  the  experiment, 
then  CT  =  A  is  the  corresponding  wave-length. 
Thus,  expressed  in  fractions  of  the  interval  .between 
the  fringes,  the  expected  displacement  becomes 
equal  to 


_ 
r  Ac2' 

By  causing  the  light  to  be  reflected  many  times 

2l 

2l  was  magnified  to  such  an  extent  that  -^  became 

A 

of     the    order    io7.       If,    for    example,    2l  =  30 
metres  =  3O-io2  cms.,  A  =  6-io~5  cms.  =  the  wave- 

2l 

length  of  sodium  light,  then  —  =  5«io7cms.     On  the 

A 

other  hand  £  is  of  the  order  (     3Q  kilometres     \« 

c2  \300,ooo  kilometres/ 

that  is,  io~  8.     The  expected  displacement  of  the 


76      THE  FOUNDATIONS  OF  EINSTEIN'S 

fringes  would  thus  have  to  be  about  0-56  of  the 
breadth  of  a  fringe.  Actually,  an  amount  of  the 
order  0-02  of  the  breadth  of  a  fringe  was  observed. 
Thus,  the  ether  wind  did  not  make  itself  remarked 
optically  in  the  motion  of  the  earth.  By  carrying 
out  the  experiment  at  different  times  of  the  year 
the  possible  objection  that  the  motion  of  translation 
of  the  entire  solar  system  might  have  counter- 
balanced the  motion  of  the  earth  in  her  orbit  was 
removed. 

The  Michelson-Morley  experiment  has  shown 
conclusively  that  there  is  no  physical  sense  in 
talking  of  absolute  rest  or  of  a  translation  relative 
to  absolute  space,  since  all  systems  that  move 
rectilinearly  and  uniformly  with  respect  to  one 
another  are  of  equal  value  for  describing  natural 
phenomena.  It  is  thus  a  matter  of  convention 
which  system  we  are  to  regard  as  at  rest  and  which 
as  being  in  motion.  We  may  assign  the  same  value 
to  the  velocity  of  light  in  all  systems.  A  detailed 
theory  of  these  fundamental  experiments  may  be 
found  in  all  comprehensive  accounts  of  the  special 
theory  of  relativity.  We  here  merely  mention  the 
original  paper  by  A.  Einstein  (Annalen  der  Physik, 
Bd.  17,  1905,  p.  891),  and  the  booklet,  "  Einfiihrung 
in  die  Relativitatstheorie,"  by  Dr.  W.  Block,  out 
of  the  series  "  Aus  Natur  und  Geisteswelt,"  Teubner, 
1918. 

Note  3  (p.  9).  Abolishing  the  transformations 
of  Newton's  principle  of  relativity  and  replacing 
them  by  the  so-called  Lorentz-Einstein  transforma- 
tions signified  a  step  of  extraordinarily  far-reaching 
consequence.  It  was  justified  in  that  the  new 
theory  of  relativity  which  followed  as  a  result  of 
it,  confirmed,  without  difficulty,  the  results  of  all 
the  fundamental  experiments  of  optics  and  electro- 
dynamics. Concerning  the  Michelson-Morley  ex- 


THEORY  OF  GRAVITATION  77 

periment,  Lorentz,  to  account  for  its  negative 
result  within  the  realm  of  electrodynamics,  had 
been  compelled  to  set  up  the  hypothesis  that  the 
dimensions  of  all  bodies  contract  in  the  direction  of 
their  motion.  But  Einstein  now  showed  that  if 
we  define  the  conception  of  simultaneity  rigorously, 
taking  into  account  the  postulate  of  the  constancy 
of  the  velocity  of  light,  the  Lorentz-transformations, 
which  had  been  found  empirically,  followed  neces- 
sarily as  those  equations  of  transformation  that 
must  hold  between  the  co-ordinates  of  two  systems 
moving  uniformly  and  rectilinearly  with  respect  to 
each  other.  And  without  the  help  of  any  further 
hypothesis  there  appears  as  a  direct  consequence  of 
this  transformation  just  that  contraction  of  lengths 
which  Lorentz  had  adduced  to  explain  the  result 
of  the  Michelson-Morley  experiment.  This  con- 
traction of  a  length  /  in  the  direction  of  motion  of 

/ v* 

an  object  to  the  value  Mi  —  _  is,  however,  in  the 

*          c 

new  theory  the  expression  of  the  general  fact  that 
the  dimensions  of  a  body  have  only  a  relative 
meaning,  that  is,  that  their  values  depend  on  the 
state  of  motion  of  the  observer,  which  determines 
the  dimensions  of  the  body  in  question.  This 
holds  for  the  extension  of  bodies  in  time  as  well 
as  in  space.  From  the  point  of  view  of  the  new 
principle  of  relativity  the  negative  result  of  the 
Michelson-Morley  experiment  was  self-explained. 
But  what  was  the  position  with  regard  to  the  other 
fundamental  facts  of  optics  and  electrodynamics  ? 
The  result  of  Fizeau's  experiment  concerning  the 
velocity  of  light  in  flowing  water  became  a  direct 
test  of  the  kinematics  arising  out  of  the  new  formulae. 
According  to  the  Lorentz  transformation  the  two 
velocities,  q  and  v  with  which,  for  example,  two 
locomotives  approach  each  other,  do  not  merely 


78       THE  FOUNDATIONS  OF  EINSTEIN'S 

become  added,  so  that  q  -f  v  would  be  the  relative 
velocity  of  each  with  respect  to  the  other,  but 
rather  each  engine-driver  will  find  as  the  velocity 
with  which  he  passes  the  other  driver,  the  value 


according  to  the  new  formulae.  This  is  the  addition 
theorem  of  velocities  according  to  the  new  theory. 
It  gives  us  immediately  the  amount  observed  in 
Fizeau's  experiment  for  the  velocity  of  light  in 
flowing  water.  Aberration  and  the  Doppler  effect 
follow  just  as  readily  to  the  correct  amount.  A 
detailed  discussion  of  these  questions  is  to  be  found 
in  every  account  of  the  "  special  "  theory  of  relativity 
(cf.  the  references  given  in  Note  2). 

Note  4  (p.  12).  Ph.  Frank  and  H.  Rothe, 
Ann.  d.  Phys.,  4  Folge,  Bd.  34,  p.  825. 

The  assumptions  for  the  general  equations  of 
transformation  by  which  two  systems  S  and  S' 
that  move  uniformly  and  rectilinearly  with  the 
velocity  q  with  respect  to  each  other  are  connected 
are  as  follows  : — 

1.  The  equations  of  transformation  form  a  linear 
homogeneous  group  in   the   variable  parameter  q. 
This  means  that  the  successive  application  of  two 
equations  of  transformation,  of  which  the  one  refers 
the  system  S  to  the  system  S',  and  the  second  S'  to 
S"  (S  is  to  have  the  constant  velocity  q  with  respect 
to  S',  and  S'  the  constant  velocity  q'  with  respect  to 
S")  again  leads  to  an  equation   of  transformation 
of  the  same  form  as  that  of  the  initial  equations. 
The  parameter  q"  that  occurs  in  the  new  equation 
depends  in  a  definite  way  on  q'  and  q. 

2.  The  contractions  of  the  lengths  depend  only 
on  the  value  of  the   parameter   q.     We   must,   of 


THEORY  OF  GRAVITATION  79 

course,  from  the  very  outset  reckon  with  the  possi- 
bility that  the  length  of  a  rod  measured  in  the 
system  that  is  at  rest  comes  out  differently  when 
measured  in  the  moving  system.  Now,  condition  2 
requires  that  if  contractions  occur  (that  is,  changes 
of  length  in  these  various  methods  of  determination) 
values  are  to  depend  only  on  the  magnitude  of  the 
velocity  of  both  systems  and  not  on  the  direction 
of  their  motion  in  space.  Thus  this  postulate 
endows  space  with  the  property  of  isotropy,  and  is 
in  fair  correspondence  with  the  postulate  of  section 
3#,  which  states  that  it  must  be  possible  to  compare 
each  line-element  with  every  other  in  length  in- 
dependently of  its  position  in  space,  and  its  direction. 

An  essential  feature  is  that  the  constancy  of  the 
velocity  of  light  is  not  demanded  in  either  of  the 
postulates  i  and  2.  Rather,  the  distinguishing  pro- 
perty of  a  definite  velocity  in  virtue  of  which  it  pre- 
serves its  value  in  all  systems  that  emerge  out  of  one 
another  through  such  transformations  is  a  direct 
corollary  to  these  two  general  postulates,  and  the 
result  of  the  Michelson-Morley  experiment  merely 
determines  the  value  of  this  special  velocity  which 
could,  of  course,  be  found  only  from  observation. 

Note  5  (p.  15).  Einstein  has  shown  in  a  simple 
example  how,  on  the  basis  of  the  formulae  of  the 
special  theory  of  relativity,  a  point-mass  loses 
inertial  mass  when  it  radiates  out  energy. 

We  assume  that  a  point-mass  emits  a  light-wave 

of  energy  -  in  a  certain  direction,  and  a  light-wave 

of  the  same  energy  --in  the  opposite    direction. 

Then,  in  view  of  the  symmetry  of  the  process  of 
emission  with  respect  to  the  system  of  reference  of 
the  co-ordinates  x,  y,  z,  t  originally  chosen,  the 
point-mass  remains  at  rest.  Let  the  total  energy 


80       THE  FOUNDATIONS  OF  EINSTEIN'S 

of  the  point-mass  be  E0  referred  to  this  system,  but 
H0  referred  to  a  second  system  which  we  suppose 
moving  with  the  uniform  velocity  v  with  respect 
to  the  first.  We  shall  apply  the  principle  of  energy 
to  this  process.  If  V  and  A  are  the  frequency 
and  amplitude  of  the  light-wave  in  the  initial 
system,  V',  A',  x',y',  z',  t'  the  frequency,  amplitude, 
and  co-ordinates  in  the  second  (the  moving)  system, 
further,  (f>  the  angle  between  the  wave-normals 
and  the  line  connecting  the  point-mass  with  the 
observer,  then  Doppler's  principle  gives  for  the 
frequency  of  the  light-wave  in  the  moving  system  : 

v 
i  —  ~  .  cos  (f> 

t  C 

"    =" 


The  formulae  of  the  special  theory  of  relativity 
give  us,  correspondingly,  for  the  amplitude  in  the 
moving  system 


I  —  -  cos  <f> 


A'  =  A  . 


According  to  Maxwell's  theory  the  energy  of  the 
light-wave  per  unit  volume  is  ^—  .  A2.  We  now  wish 

O7T 

to  calculate  the  corresponding  energy-density  also 
with  respect  to  the  moving  system.  We  must  here 
take  into  account  that,  in  consequence  of  the  con- 
traction of  the  lengths  according  to  the  Lorentz- 
Einstein  transformation  formulae,  the  volume  V  of  a 
sphere  in  the  resting  system  becomes  transformed 
into  that  of  an  ellipsoid  as  measured  from  the 
moving  system  V ;  indeed,  this  volume  of  the  ellip- 
soid is 


THEORY  OF  GRAVITATION 


81 


V'  =  V. 


V  , 

-COS  </> 

c 


Hence  the  energy-densities  in  the  accented  and  un- 
accented system  are  in  the  ratio  : 


U 
L 


A'2.V 

Sir 


877 


.A2.V 


If  we  now  designate  the  energy-content  of  the 
point-mass  after  the  emission  by  Ex,  and  the  cor- 
responding quantity  referred  to  the  moving  system 
by  H!,  then  we  have  : 


whereas 


/  D2      +    2         /  V* 

V1-^       V1-^ 


From  this  we  get  directly  that 
[H0  -  E0]  -  [H,  -  EJ=  L 


What  does  this  equation  assert  ? 

H  and  E  are  the  energy- values  of  the  same  point- 
mass,  in  the  first  place  referred  to  a  system  with 
respect  to  which  the  point-mass  moves,  and  in  the 
second  related  to  a  system  in  which  the  point-mass 
6 


82      THE  FOUNDATIONS  OF  EINSTEIN'S 

is  at  rest.  Hence  the  difference  H  —  E,  except 
for  an  additive  constant,  must  be  equal  to  the 
kinetic  energy  of  the  point-mass  referred  to  the 
moving  system.  Thus,  we  may  write 

H0  -  E0  =  K0  +  C          Hx  -  E!  =  Kx  +  C 

wherein  C  denotes  a  constant  which  does  not  alter 
during  the  light-emission  of  the  point-mass,  since, 
owing  to  the  symmetry  of  the  process,  the  point- 
mass  remains  at  rest  with  respect  to  the  initial 
system.  So  we  arrive  at  the  relation  : 


In  words  this  equation  states  that  owing  to  the 
point-mass  emitting  the  energy  L  as  light,  its 
kinetic  energy  referred  to  a  moving  system  sinks 
from  the  value  K0  to  the  value  K,  corresponding  to 

a  loss  in  inertial  mass  of  the  amount  — .     For,  accor- 

c2 

ding  to  classical  mechanics,  the  expression  \  m  .  vz 
in  which  m  is  the  inertial  mass  of  the  observed 
body,  is  a  measure  of  the  kinetic  energy  of  this 
body  referred  to  a  system  with  respect  to  which 

it  moves  with  the  velocity  v.    Thus  —  must  be 

c 

taken  as  standing  for  the  inertial  mass  of  an  amount 
of  energy  L. 

Note  6  (p.  29).  The  facts  that  every  pair  of  points 
(point-pair)  in  space  have  the  same  magnitude- 
relation  (viz.  the  same  expression  for  the  mutual 
distance  between  them)  and  that  with  the  aid  of 
this  relation,  every  point-pair  can  be  compared 
with  every  other,  constitute  the  characteristic 
feature  which  distinguishes  space  from  the  remaining 


THEORY  OF  GRAVITATION  83 

continuous  manifolds  which  are  known  to  us.  We 
measure  the  mutual  distance  between  two  points  on 
the  floor  of  a  room,  and  the  mutual  distance  between 
two  points  which  lie  vertically  above  one  another 
on  the  wall,  with  the  same  measuring-scale,  which 
we  thus  apply  in  any  direction  at  pleasure.  This 
enables  us  to  "  compare  "  the  mutual  distance  of 
a  point-pair  on  the  floor  with  the  mutual  distance 
of  any  other  pair  of  points  on  the  wall. 

In  the  system  of  tones,  on  the  contrary,  quite 
different  conditions  prevail.  The  system  of  tones 
represents  a  manifold  of  two  dimensions,  if  one 
distinguishes  every  tone  from  the  remaining  tones 
by  its  pitch  and  its  intensity.  It  is,  however,  not 
possible  to  compare  the  "  distance  "  between  two 
tones  of  the  same  pitch  but  different  intensity  (ana- 
logous to  the  two  points  on  the  floor)  with  the 
"  distance  "  between  two  tones  of  different  pitch 
but  equal  intensity  (analogous  to  the  two  points  on 
the  wall).  The  measure-conditions  are  thus  quite 
different  in  this  manifold. 

In  the  system  of  colours,  too,  the  measure-relations 
have  their  own  peculiarity.  The  dimensions  of  the 
manifold  of  colours  are  the  same  as  those  of  space, 
as  each  colour  can  be  produced  by  mixing  the 
three  "  primary  "  colours.  But  there  is  no  relation 
between  two  arbitrary  colours,  which  would  corre- 
spond to  the  distance  between  two  points  in 
space.  Only  when  a  third  colour  is  derived  by 
mixing  these  two,  does  one  obtain  an  equation 
between  these  three  colours  similar  to  that  which 
connects  three  points  in  space  lying  in  one  straight 
line. 

These  examples,  which  are  borrowed  from 
Helmholtz's  essays,  serve  to  show  that  the  measure- 
relations  of  a  continuous  manifold  are  not  already 
given  in  its  definition  as  a  continuous  manifold, 


84      THE  FOUNDATIONS  OF  EINSTEIN'S 

nor  by  fixing  its  dimensions.  A  continuous  mani- 
fold generally  allows  of  various  measure-relations. 
It  is  only  experience  which  enables  us  to  derive 
the  measure-laws  which  are  valid  for  each  particular 
manifold.  The  fact,  discovered  by  experience, 
that  the  dimensions  of  bodies  are  independent  of 
their  particular  position  and  motion,  led  to  the 
laws  of  Euclidean  geometry  where  congruence  is  the 
deciding  factor  in  comparing  various  portions  of 
space.  These  questions  have  been  exhaustively 
treated  by  Helmholtz  in  various  essays.  Refer- 
ences : — 

Riemann,  "  Uber  die  Hypothesen,  welche  der 
Geometric  zugrunde  liegen  "  (1854).  Newly  pub- 
lished and  annotated  by  H.  Weyl,  Berlin,  1919. 

Helmholtz.  "  Ueber  die  tatsachlichen  Grund- 
lagen  der  Geometric,"  Wiss.  Abh.  2,  S.  10. 

Helmholtz.  "  Ueber  die  Tatsachen,  welche  der 
Geometrie  zugrunde  liegen,"  Wiss.  Abh.  2,  S.  618. 

Helmholtz.  "  Ueber  den  Ursprung  und  die  Be- 
deutung  der  geometrischen  Axiome,"  Vortrage  und 
Reden,  Bd.  2,  S.  i. 

Note  7  (p.  26).  The  postulate  that  finite  rigid 
bodies  are  to  be  capable  of  free  motions,  can  be 
most  strikingly  illustrated  in  the  realm  of  two- 
dimensions.  Let  us  imagine  a  triangle  to  be  drawn 
upon  a  sphere,  and  also  upon  a  plane :  the  former 
being  bounded  by  arcs  of  great  circles  and  the 
latter  by  straight  lines  ;  one  can  then  slide  these 
triangles  over  their  respective  surfaces  at  will,  and 
can  make  them  coincide  with  other  triangles, 
without  thereby  altering  the  lengths  of  the  sides 
or  the  angles.  Gauss  has  shown  that  this  is  possible 
because  the  curvature  at  every  point  of  the  sphere 
(or  the  plane,  respectively)  has  exactly  the  same 
value.  And  yet  the  geometry  of  curves  traced 
upon  a  sphere  is  different  from  that  of  curves  traced 


THEORY  OF  GRAVITATION  85 

upon  a  plane,  for  the  reason  that  these  two  con- 
figurations cannot  be  deformed  into  one  another 
without  tearing  (vide  Note  27).  But  upon  both  of 
them  planimetrical  figures  can  be  freely  shifted 
about,  and,  therefore,  theorems  of  congruence  hold 
upon  them.  If,  however,  we  were  to  define  a  cur- 
vilinear triangle  upon  an  egg-shaped  surface  by 
the  three  shortest  lines  connecting  three  given 
points  upon  it,  we  should  find  that  triangles  could 
be  constructed  at  different  places  on  this  surface, 
having  the  same  lengths  for  the  sides  ;  but  these 
sides  would  enclose  angles  different  from  those 
included  by  the  corresponding  sides  of  the  initial 
triangle,^  and,  consequently,  such  triangles  would 
not  be  congruent,  in  spite  of  the  fact  that  corre- 
sponding sides  are  equal.  Figures  upon  an  egg- 
shaped  surface  cannot,  therefore,  be  made  to  slide 
over  the  surface  without  altering  their  dimensions  : 
and  in  studying  the  geometrical  conditions  upon 
such  a  surface,  we  do  not  arrive  at  the  usual  theorems 
of  congruence.  Quite  analogous  arguments  can 
be  applied  to  three-  and  four-dimensional  realms  : 
but  the  latter  cases  offer  no  corresponding  pictures 
to  the  mind.  If  we  demand  that  bodies  are  to  be 
freely  movable  in  space  without  suffering  a  change 
of  dimensions,  the  "  curvature  "  of  the  space  must 
be  the  same  at  every  point.  The  conception  of 
curvature,  as  applied  to  any  manifold  of  more  than 
two  dimensions,  allows  of  strict  mathematical 
formulation ;  the  term  itself  only  hints  at  its 
analogous  meaning,  as  compared  with  the  conception 
of  curvature  of  a  surface.  In  three-dimensional 
space,  too,  various  cases  can  be  distinguished, 
similarly  to  plane-  and  spherical-geometry  in  two- 
dimensional  space.  Corresponding  to  the  sphere, 
we  have  a  non-Euclidean  space  with  constant 
positive  curvature  ;  corresponding  to  the  plane  we 


86      THE  FOUNDATIONS  OF  EINSTEIN'S 

have  Euclidean  space  with  curvature  zero.  In  both 
these  spaces  bodies  can  be  moved  about  without 
their  dimensions  altering  ;  but  Euclidean  space  is 
furthermore  infinitely  extended  :  whereas  "  spher- 
ical "  space,  though  unbounded,  like  the  surface 
of  a  sphere,  is  not  infinitely  extended.  These 
questions  are  to  be  found  extensively  treated  in 
a  very  attractive  fashion  in  Helmholtz's  familiar 
essay :  "  Ueber  den  Ursprung  und  die  Bedeutung 
der  geometrischen  Axiome  "  (Vortrdge  und  Reden, 
Bd.  2,  S.  i). 

Note  8  (p.  26).  The  properties,  which  the  ana- 
lytical expression  for  the  length  of  the  line-element 
must  have,  may  be  understood  from  the  following : 

Let  the  numbers  xlt  x2  denote  any  point  of  any 
continuous  two-dimensional  manifold,  e.g.  a  surface. 
Then,  together  with  this  point,  a  certain  "  domain  " 
around  the  point  is  given,  which  includes  points  all 
of  which  lie  in  the  plane. — D.  Hilbert  has  strictly 
defined  the  conception  of  a  multiply-extended 
magnitude  (i.e.  a  manifold)  upon  the  basis  of  the 
theory  of  aggregates  in  his  "  Grundlagen  der  Geo- 
metric "  (p.  177).  In  this  definition  the  conception 
of  the  "  domain  "  encircling  a  point  is  made  to  give 
Riemann's  postulate  of  the  continuous  connection 
existing  between  the  elements  of  a  manifold  and  a 
strict  form. 

Setting  out  from  the  point  xlt  x2  we  can  contin- 
uously pass  into  its  domain,  and  at  any  point,  e.g. 
#1  +  <foi»  xz  +  dxz>  inquire  as  to  the  "  distance  "  of 
this  point  from  the  starting-point.  The  function 
which  measures  this  distance  will  depend  upon  the 
values  of  xlt  xz,  dxlt  dx2,  and  for  every  intermediate 
point  of  the  path  which  has  conducted  us  from  xlt  x2 
to  the  point  x±  -f-  dxlf  x2  +*dx2  will  successively 
assume  certain  continually  changing,  and,  as  we 
may  suppose,  continually  increasing,  values.  At 


THEORY  OF  GRAVITATION  87 

the  point  xlt  x2  itself  it  will  assume  the  value  zero, 
and  for  every  other  point  of  the  domain  its  value 
must  be  positive.  Moreover,  we  shall  expect  to 
find  that,  for  any  intermediate  point,  denoted  by 

required  function  which  measures  the  distance  of 
this  point  from  xlf  x2,  will,  at  this  point,  have  a 
value  half  that  of  its  value  for  the  point  xt  +  dxlt 
x2  -f-  dx2.  Under  these  assumptions,  the  function 
will  be  homogeneous  and  of  the  first  degree  in  the 
dx's  ;  its  value  will  then  appear  multiplied  by  that 
factor  in  proportion  to  which  the  dx's  were  increased. 
In  addition,  it  must  itself  vanish  if  all  the  dx's  are 
zero  ;  and  if  they  all  change  their  sign  it  must  not 
alter  its  value,  which  always  remains  positive.  It 
will  immediately  be  evident  that  the  function 

ds  =  t/gudxf  +  gl2dx^x2  -f  g22dx22 


fulfils  all  these  requirements  ;  but  it  is  by  no  means 
the  only  function  of  this  kind. 

Note  9  (p.  29).  But  the  expression  of  the  fourth 
degree  for  the  line  element  would  not  permit  of  any 
geometrical  interpretation  of  the  formula,  such  as 
is  possible  with  the  expression 


which  latter  may  be  regarded  as  a  general  case  of 
Pythagoras'  theorem. 

'Note  10  (p.  30).  By  a  "discrete"  manifold 
we  mean  one  in  which  no  continuous  transition  of 
the  single  elements  from  one  to  another  is  possible, 
but  each  element  to  a  certain  extent  represents  an 
independent  entity.  The  aggregate  of  all  whole 
numbers,  for  instance,  is  a  manifold  of  this  type, 
or  the  aggregate  of  all  planets  in  our  solar  system, 
etc.,  and  many  other  examples  may  be  found  ;  and 


88       THE  FOUNDATIONS  OF  EINSTEIN'S 

indeed  all  finite  aggregates  in  the  theory  of  aggre- 
gates are  such  discrete  manifolds.  "  Measuring," 
in  the  case  of  discrete  manifolds,  is  performed 
merely  by  "  counting,"  and  does  not  present  any 
special  difficulties,  as  all  manifolds  of  this  type 
are  subject  to  the  same  principle  of  measurement. 
When  Riemann  then  proceeds  to  say :  "  Either, 
therefore,  the  reality  which  underlies  space  must 
form  a  discrete  manifold,  or  we  must  seek  the  ground 
of  its  metric  relations  outside  it,  in  binding  forces 
which  act  upon  it,"  he  only  wishes  to  hint  at  a 
possibility,  which  is  at  present  still  remote,  but 
which  must,  in  principle,  always  be  left  open.  In 
just  the  last  few  years  a  similar  change  of  view 
has  actually  occurred  in  the  case  of  another  mani- 
fold which  plays  a  very  important  part  in  physics, 
viz.  "  energy  "  ;  the  meaning  of  the  hint  Riemann 
gives  will  become  clearer  if  we  consider  this 
example. 

Up  till  a  few  years  ago,  the  energy  which  a  body 
emanates  by  radiation  was  regarded  as  a  contin- 
uously variable  quantity :  and  attempts  were  there- 
fore made  to  measure  its  amount  at  any  particular 
moment  by  means  of  a  continuously  varying 
sequence  of  numbers.  The  researches  of  Max 
Planck  have,  however,  led  to  the  view  that  this 
energy  is  emitted  in  "  quanta,"  and  that  therefore 
the  "  measuring  "  of  its  amount  is  performed  by 
counting  the  number  of  "  quanta."  The  reality 
underlying  radiant  energy,  according  to  this,  is  a 
discrete  and  not  a  continuous  manifold.  If  we  now 
suppose  that  the  view  were  gradually  to  take  root 
that,  on  the  one  hand,  all  measurements  in  space 
only  have  to  do  with  distances  between  ether-atoms  ; 
and  that,  on  the  other  hand,  the  distances  of  single 
ether-atoms  from  one  another  can  only  assume 
certain  definite  values,  all  distances  in  space  would 


THEORY  OF  GRAVITATION  89 

be  obtained  by  "  counting  "  these  values,  and  we 
should  have  to  regard  space  as  a  discrete  manifold. 

Note  ii  (p.  32).  C.  Neumann.  "  Ueber  die 
Prinzipien  der  Galilei-Newtonschen  Theorie,"  Leipzig 
1870,  S.  18. 

Note  12  (p.  32).  H.  Streintz.  "  Die  physikali- 
schen  Grundlagen  der  Mechanik,"  Leipzig,  1883. 

Note  13  (p.  33).  A.  Einstein.  "  Annalen  der 
Physik,"  4  Folge,  Bd.  17,  S.  891. 

Note  14  (p.  35).  Minkowski  was  the  first  to 
call  particular  attention  to  this  deduction  of  the 
special  principle  of  relativity. 

Note  15  (p,  38).  The  term  "  inertial  system  " 
was  originally  not  associated  with  the  system,  which 
Neumann  attached  to  the  hypothetical  body  Alpha. 
Nowadays  it  is  generally  understood  to  signify  a 
rectilinear  system  of  co-ordinates,  relatively  to  which 
a  point-mass,  which  is  only  subject  to  its  own 
inertia,  moves  uniformly  in  a  straight  line.  Whereas 
C.  Neumann  only  invented  the  body  Alpha,  as  an 
absolutely  hypothetical  configuration,  in  order  to 
be  able  to  formulate  the  law  of  inertia,  later  re- 
searches, especially  those  of  Lange,  tended  to  show 
that,  on  the  basis  of  rigorous  kinematical  considera- 
tions, a  co-ordinate  system  could  be  derived,  which 
would  possess  the  properties  of  such  an  inertial 
system.  However,  as  C.  Neumann  and  J.  Petzoldt 
have  demonstrated,  these  developments  contain 
faulty  assumptions,  and  give  the  law  of  inertia  no 
firmer  basis  than  the  body  Alpha  introduced  by 
Neumann. 

Such  an  inertial  system  is  determined  by  the 
straight  lines  which  connect  three  point-masses 
infinitely  distant  from  one  another  (and  thus  unable 
to  exert  a  mutual  influence  upon  one  another)  and 
which  are  not  subject  to  any  other  forces.  This 
definition  makes  it  evident  why  no  inertial  system 


90      THE  FOUNDATIONS  OF  EINSTEIN'S 

will  be  discoverable  in  nature,  and  why,  consequently, 
the  law  of  inertia  will  never  be  able  to  be  formulated 
so  as  to  satisfy  the  physicist.  References  : — 

C.  Neumann.  "  Ueber  die  Prinzipien  der  Galilei- 
Newtonschen  Theorie,"  Leipzig,  1870. 

L.  Lange.  "  Berichte  der  Kgl.  Sachs.  Ges.  d. 
Wissenschaften.  Math.-phil.  Klasse,"  1885. 

L.  Lange.  "  Die  Geschichte  der  Entwickelung 
des  Bewegungsbegriffes,"  Leipzig,  1886. 

H.  Seeliger.  "  Ber.  der  Bayr.  Akademie,"  1906, 
Heft  i. 

C.  Neumann.  "  Ber  der  Kgl.  Sachs.  Ges.  d.  Wiss. 
Math.-phys.  Klasse,"  1910,  Bd.  62,  S.  69  and  383. 

J.  Petzoldt.     "  Ann.  der  Naturphilosophie,"  Bd.  7. 

Note  1 6  (p.  38).  E.  Mach.  "  Die  Mechanik  in 
in  ihrer  Entwickelung,"  4  Aufl.  S.  244. 

Note  17  (p.  40).  The  new  points  of  view  as 
to  the  nature  of  inertia  are  based  upon  the  study 
of  the  electromagnetic  phenomena  of  radiation. 
The  special  theory  of  relativity,  by  stating  the 
theorem  of  the  inertia  of  energy,  organically  grafted 
these  views  on  to  the  existing  structure  of  theo- 
retical physics.  The  dynamics  of  cavity-radiation, 
i.e.  the  dynamics  of  a  space  enclosed  by  walls 
without  mass,  and  filled  with  electromagnetic 
radiation,  taught  us  that  a  system  of  this  kind 
opposes  a  resistance  to  every  change  of  its  motion, 
just  like  a  heavy  body  in  motion.  The  study  of 
electrons  (free  electric  charges)  in  a  state  of  free 
motion,  e.g.  in  a  cathode-tube,  taught  us  likewise 
that  these  exceedingly  small  particles  behave  like 
inert  bodies  ;  that  their  inertia  is  not,  however, 
conditioned  by  the  matter  to  which  they  might 
happen  to  be  attached,  but  rather  by  the  electro- 
magnetic effects  of  the  field  to  which  the  moving 
electron  is  subject.  This  gave  rise  to  the  concep- 
tion of  the  apparent  (electromagnetic)  mass  of  an 


THEORY  OF  GRAVITATION  91 

electron.  The  special  theory  of  relativity  finally 
led  to  the  conclusion  that  to  all  energy  must  be 
accorded  the  property  of  inertia. 

Every  body  contains  energy  (e.g.  a  certain  definite 
amount  in  the  form  of  heat-radiation  internally). 
The  inertia,  which  the  body  reveals,  is  thus  partly 
to  be  debited  to  the  account  of  this  contained 
energy.  As  this  share  of  inertia  is,  according  to 
the  special  theory  of  relativity,  relative  (i.e.  re- 
presents a  quantity  which  depends  upon  the  choice 
of  the  system  of  reference),  the  whole  amount  of 
the  inertial  mass  of  the  body  has  no  absolute  value, 
but  only  a  relative  one.  This  energy-content  of 
radiant  heat  is  distributed  throughout  the  whole 
volume  of  each  particular  body ;  one  can  thus 
speak  of  the  energy-content  of  unit  volume.  This 
enables  us  to  derive  the  notion  of  density  of  energy. 
The  density  of  the  energy  (i.e.  amount  per  unit 
volume)  is  thus  a  quantity,  the  value  of  which  is 
also  dependent  upon  the  system  of  reference. 
References  : — 

M.  Planck.     "  Ann.  der  Phys.,"  4  Folge,  Bd.  26. 

M.  Abraham.  "  Elect romagnetische  Energie  der 
Strahlung,"  4  Aufl.,  1908. 

Note  1 8  (p.  40).  The  determination  of  the  inertial 
mass  of  a  body  by  measuring  its^weight  is  rendered 
possible  only  by  the  experimental  fact  that  all 
bodies  fall  with  equal  acceleration  in  the  gravita- 
tional field  at  the  earth's  surface.  If  p  and  p' 
denote  the  pressures  of  two  bodies  upon  the  same 
support  (i.e.  their  respective  weights),  and  g  denote 
the  acceleration  due  to  the  earth's  gravitational  field 
at  the  point  in  question,  then  p  =  m  .  g  dynes  and 
p'  =  m' .  g  dynes,  respectively,  where  m  and  m' 
are  the  factors  of  proportionality,  and  are  called 
the  masses  of  the  two  bodies,  respectively.  As  g  has 
the  same  value  in  both  equations,  we  have 


92      THE  FOUNDATIONS  OF  EINSTEIN'S 


p        m 

and  we  can  accordingly  measure  the  masses  of  two 
bodies  at  the  same  place,  by  determining  their 
weights. 

Although  Galilei  and  Newton  had  already  known 
that  all  bodies  at  the  same  place  fall  with  the  same 
velocity  (if  the  resistance  of  the  air  be  eliminated), 
this  very  remarkable  fact  has  not  received  any 
recognition  in  the  foundations  of  mechanics.  Ein- 
stein's principle  of  equivalence  is  the  first  to  assign 
to  it  the  position  to  which  it  is,  beyond  doubt, 
entitled. 

Note  19  (p.  41).  Arguing  along  the  same  lines 
B.  and  J.  Friedlander  have  suggested  an  ex- 
periment to  show  the  relativity  of  rotational 
motions,  and,  accordingly,  the  reversibility  of 
centrifugal  phenomena  (*'  Absolute  and  Relative 
Motion/'  Berlin,  Leonhard  Simion,  1896).  On 
account  of  the  smallness  of  the  effect,  the  experiment 
cannot,  at  present,  be  performed  successfully  ; 
but  it  is  quite  appropriate  for  making  the  physical 
content  of  this  postulate  more  evident.  The  fol- 
lowing remarks  may  be  quoted  : 

"  The  torsion-balance  is  the  most  sensitive  of 
all  instruments.  The  largest  rotating-masses,  with 
which  we  can  experiment,  are  probably  the  large 
fly-wheels  in  rolling-mills  and  other  big  factories. 
The  centrifugal  forces  assert  themselves  as  a  pressure 
which  tends  from  the  axis  of  rotation.  If,  therefore, 
we  set  up  a  torsion-balance  in  somewhat  close 
proximity  to  one  of  these  large  fly-wheels,  in  such  a 
position  that  the  point  of  suspension  of  the  movable 
part  of  the  torsion-balance  (the  needle)  lies  exactly, 
or  as  nearly  as  possible,  in  the  continuation  of  the 
axis  of  the  fly-wheel,  the  needle  should  endeavour 


THEORY  OF  GRAVITATION  93 

to  set  itself  parallel  to  the  plane  of  the  fly-wheel, 
if  it  is  not  originally  so,  and  should  register  a  corre- 
sponding displacement.  For  centrifugal  force  acts 
upon  every  portion  of  mass  which  does  not  lie  exactly 
in  the  axis  of  rotation,  in  such  a  way  as  to  tend  to 
increase  the  distance  of  the  mass  from  the  axis. 
It  is  immediately  apparent  that  the  greatest  possible 
displacement-effect  is  attained  when  the  needle  is 
parallel  to  the  plane  of  the  wheel." 

This  proposed  experiment  of  B.  and  J.  Friedlander 
is  only  a  variation  of  the  experiment  which  per- 
suaded Newton  to  his  view  of  the  absolute  character 
of  rotation.  Newton  suspended  a  cylindrical  vessel 
filled  with  water  by  a  thread,  and  turned  it  about 
the  axis  denned  by  the  thread  till  the  thread  became 
quite  stiff.  After  the  vessel  and  the  contained 
liquid  had  completely  come  to  rest,  he  allowed  the 
thread  to  untwist  itself  again,  whereby  the  vessel 
and  the  liquid  started  to  rotate  rapidly.  He  thereby 
made  the  following  observations.  Immediately  after 
its  release  the  vessel  alone  assumed  a  motion  of  rota- 
tion, since  the  friction  (viscosity)  of  the  water  was 
not  sufficient  to  transmit  the  rotation  immediately 
to  the  water.  So  long  as  this  state  of  affairs  pre- 
vailed, the  surface  of  the  water  remained  a  hori- 
zontal plane.  But  the  more  rapidly  the  water  was 
carried  along  by  the  rotating  walls  of  the  vessel,  the 
more  definitely  did  the  centrifugal  forces  assert 
themselves,  and  drive  the  water  up  the  walls,  so 
that  finally  its  free  surface  assumed  the  form  of  a 
paraboloid  of  revolution.  From  these  observations 
Newton  concluded  that  the  rotation  of  the  walls 
of  the  vessel  relative  to  the  water  does  not  call  up 
forces  in  the  latter.  Only  when  the  water  itself 
shares  in  the  rotation,  do  the  centrifugal  forces 
make  their  appearance.  From  this  he  came  to  his 
conclusion  of  the  absolute  character  of  rotations. 


94       THE  FOUNDATIONS  OF  EINSTEIN'S 

This  experiment  became  a  subject  of  frequent 
discussion  later  :  and  E.  Mach  long  ago  objected 
to  Newton's  deduction,  and  pointed  out  that  it 
cannot  be  straightway  affirmed  that  the  rotation 
of  the  walls  of  the  vessel  relative  to  the  water  is 
entirely  without  effect  upon  the  latter.  He  regards 
it  as  quite  conceivable  that,  provided  the  mass  of 
the  vessel  were  large  enough,  e.g.  if  its  walls 
were  many  kilometres  thick,  then  the  free  surface 
of  the  water  which  is  at  rest  in  the  rotating  vessel 
would  not  remain  plane.  This  objection  is  quite 
in  keeping  with  the  view  entailed  by  the  general 
theory  of  relativity.  According  to  the  latter,  the 
centrifugal  forces  can  also  be  regarded  as  gravita- 
tional forces,  which  the  total  sum  of  the  masses 
rotating  around  the  water  exerts  upon  it.  The 
gravitational  effect  of  the  walls  of  the  vessel  upon 
the  enclosed  liquid  is,  of  course,  vanishingly  small 
compared  with  that  of  all  the  masses  in  the  universe. 
It  is  only  when  the  water  is  in  rotation  relatively 
to  all  these  masses  that  perceptible  centrifugal 
forces  are  to  be  expected.  The  experiment  of  B. 
and  J.  Friedlander  was  intended  to  refine  the  ex- 
periment performed  by  Newton,  by  using  a  sensitive 
torsion-balance  susceptible  to  exceedingly  small 
forces  in  place  of  the  water,  and  by  substituting  a 
huge  fly-wheel  for  the  vessel  which  contained  the 
water.  But  this  arrangement,  too,  can  lead  to 
no  positive  result,  as  even  the  greatest  fly-wheel  at 
present  available  represents  only  a  vanishingly 
small  mass  compared  with  the  sum-total  of  masses 
in  the  universe. 

Note  20  (p.  42).  We  use  the  term  "  field  of 
force  "  to  denote  a  field  in  which  the  force  in  question 
varies  continuously  from  place  to  place,  and  is  given 
for  each  point  in  the  field  by  the  value  of  some 
function  of  the  place.  The  centrifugal  forces  in  the 


THEORY  OF  GRAVITATION  95 

interior  and  on  the  outer  surface  of  a  rotating  body 
are  so  distributed  as  to  compose  a  field  of  this  kind 
throughout  the  whole  volume  of  the  body,  and  there 
is  nothing  to  hinder  us  from  imagining  this  field 
to  extend  outwards  beyond  the  outer  surface  of 
the  body,  e.g.  beyond  the  surface  of  the  earth  into 
its  own  atmosphere.  We  can  thus  briefly  speak  of 
the  whole  field  as  the  centrifugal  field  of  the  earth  ; 
and,  as  the  centrifugal  field,  according  to  the  older 
views,  is  conditioned  only  by  the  inertia  of  bodies, 
and  not  by  their  gravitation,  we  can  further  speak 
of  it  as  an  inertia!  field,  in  centra-distinction  to 
the  gravitational  field,  under  the  influence  of  which 
all  bodies  which  are  not  suspended  or  supported 
fall  to  earth. 

Accordingly  the  effects  of  various  fields  of  force 
are  superposed  at  the  earth's  surface  :  (i)  the  effect 
of  the  gravitational  field,  due  to  the  gravitation  of 
the  particles  of  the  earth's  mass  towards  one  another, 
and  which  is  directed  towards  the  centre  of  the  earth  ; 
(2)  the  effect  of  the  centrifugal  field,  which,  accord- 
ing to  Einstein's  view,  can  be  regarded  as  a  gravita- 
tional field,  and  the  direction  of  action  of  which  is 
outwards  and  parallel  to  the  plane  of  the  meridian 
of  latitude  ;  finally  (3)  the  effect  of  the  gravitational 
field,  due  to  the  various  heavenly  bodies,  foremost 
amongst  them,  the  sun  and  the  moon. 

Note  21  (p.  42).  Eotvos  has  published  the  re- 
sults of  his  measurements  in  the  "  Mathematische 
und  Naturwissenschaftliche  Berichte  aus  Ungarn," 
Bd.  8,  S.  64,  1891.  A  detailed  account  is  given  by 
D.  Pekar,  "  Das  Gesetz  der  Proportionality  von 
Tragheit  und  Gravitation."  "Die  Naturwissen- 
schaft,"  1919,  7,  p.  327. 

Whereas  the  earlier  investigations  of  Newton  and 
Bessel  ("  Astr.  Nachr."  10,  S.  97,  and  "  Abhand- 
lungen  von  Bessel,"  Bd.  3,  S.  217),  about  the 


96        THE  FOUNDATIONS  OF  EINSTEIN'S 

attractive  effect  of  the  earth  upon  various  sub- 
stances, are  based  upon  observations  with  a  pendu- 
lum, Eotvos  worked  with  sensitive  torsion-balances. 

The  force,  in  consequence  of  which  bodies  fall,  is 
composed  of  two  components  :  first  the  attractive 
force  of  the  earth,  which  (except  for  deviations 
which  may,  for  the  present,  be  neglected)  is  directed 
towards  the  centre  of  the  earth  ;  and,  second,  the 
centrifugal  force,  which  is  directed  outwards  parallel 
to  the  meridians  of  latitude.  If  the  attraction  of 
the  earth  upon  two  bodies  of  equal  mass  but  of 
different  substance  were  different,  the  resultant  of 
the  attractive  and  centrifugal  forces  would  point  in 
a  different  direction  for  each  body.  Eotvos  then 
states:  "By  calculation  we  find  that  if  the  attractive 
effect  of  the  earth  upon  two  bodies  of  equal  mass, 
but  composed  of  different  substance,  differed  by 
a  thousandth,  the  directions  of  the  gravitational 
forces  acting  upon  the  two  bodies  respectively 
would  make  an  angle  of  0-356  second,  i.e.  about 
a  third  of  a  second  with  one  another  ;  "  and  if  the 
difference  in  the  attractive  force  were  to  amount 
to  a  twenty-millionth,  this  angle  would  have  to  be 
seconds  ;  that  is,  slightly  more  than 
of  a  second  ;  and  later  : 

"  I  attached  separate  bodies  of  about  30  grms. 
weight  to  the  end  of  a  balance-beam  about  25  to 
30  cms.  long,  suspended  by  a  thin  platinum  wire 
in  my  torsion-balance.  After  the  beam  had  been 
placed  in  a  position  perpendicular  to  the  meridian, 
I  determined  its  position  exactly  by  means  of  two 
mirrors,  one  fixed  to  it  and  another  fastened  to  the 
case  of  the  instrument.  I  then  turned  the  instru- 
ment, together  with  the  case,  through  180°,  so  that 
the  body  which  was  originally  at  the  east  end  of 
the  beam  now  arrived  at  the  west  end  :  I  then  de- 
termined the  position  of  the  beam  again,  relative 


THEORY  OF  GRAVITATION  97 

to  the  instrument.  If  the  resultant  weights  of  the 
bodies  attached  to  both  sides  pointed  in  different 
directions,  a  torsion  of  the  suspending  wire  should 
ensue.  But  this  did  not  occur  in  the  cases  in  which 
a  brass  sphere  was  constantly  attached  to  the  one 
side,  and  glass,  cork,  or  crystal  antimony  was 
attached  to  the  other ;  and  yet  a  deviation  of 
^.^ooth  of  a  second  in  the  direction  of  the 
gravitational  force  would  have  produced  a  torsion 
of  one  minute,  and  this  would  have  been  observed 
accurately." 

Eotvos  thus  attained  a  degree  of  accuracy,  such 
as  is  approximately  reached  in  weighing ;  and  this 
was  his  aim  :  for  his  method  of  determining  the 
mass  of  bodies  by  weighing  is  founded  upon  the 
axiom  that  the  attraction  exerted  by  the  earth 
upon  various  bodies  depends  only  upon  their  mass, 
and  not  upon  the  substance  composing  them.  This 
axiom  had,  therefore,  to  be  verified  with  the  same 
degree  of  accuracy  as  is  attained  in  weighing.  If  a 
difference  of  this  kind  in  the  gravitation  of  various 
bodies  having  the  same  mass  but  being  composed  of 
different  substance  exists  at  all,  it  is,  according  to 
Eotvos,  less  than  a  twenty-millionth  for  brass,  glass, 
antimonite,  cork,  and  less  than  a  hundred-thousandth 
for  air. 

Note  22  (p.  44).  Vide  also  A.  Einstein,  "  Grund- 
lagen  der  allgemeinen  Relativitatstheorie,"  "Ann.  d. 
Phys.,"  4  Folge,  Bd.  49,  S.  769. 

Note  23  (p.  46).  The  equation  8{$ds}  =  o  asserts 
that  the  variation  in  the  length  of  path  between 
two  sufficiently  near  points  of  the  path  vanishes 
for  the  path  actually  traversed ;  i.e.  the  path 
actually  chosen  between  two  such  points  is  the 
shortest  of  all  possible  ones.  If  we  retain  the  view 
of  classical  mechanics  for  a  moment,  the  following 
example  will  give  us  the  sense  of  the  principle 
7 


98       THE  FOUNDATIONS  OF  EINSTEIN'S 

clearly :  In  the  case  of  the  motion  of  a  point- 
mass,  free  to  move  about  in  space,  the  straight 
line  is  always  the  shortest  connecting  line  between 
two  points  in  space  :  and  the  point-mass  will  move 
from  the  one  point  to  the  other  along  this  straight 
line,  provided  no  other  disturbing  influences  come 
into  play  (Law  of  inertia).  If  the  point-mass  is 
constrained  to  move  over  any  curved  surface,  it 
will  pass  from  one  point  to  another  along  a  geodetic 
line  to  the  surface,  since  the  geodetic  lines  represent 
the  shortest  connecting  lines  between  points  on  the 
surface.  In  Einstein's  theory  there  is  a  fully  corre- 
sponding principle,  but  of  a  much  more  general  form. 
Under  the  influence  of  inertia  and  gravitation  every 
point-mass  passes  along  the  geodetic  lines  of  the 
space-time-manifold.  The  fact  of  these  lines  not, 
in  general,  being  straight  lines,  is  due  to  the  gravi- 
tational field,  in  a  certain  sense,  putting  the  point- 
mass  under  a  sort  of  constraint,  similar  to  that 
imposed  upon  the  freedom  of  motion  of  the  point- 
mass  by  a  curved  surface.  A  principle  in  every 
way  corresponding  had  already  been  installed  in 
mechanics  as  a  fundamental  principle  for  all  motions 
by  Heinrich  Hertz. 

Note  24  (p.  48).  Vide  A.  Einstein,  "Ann.  d. 
Phys.,"  4  Folge,  Bd.  35,  S.  898. 

Note  25  (p.  48).  The  expression  "  accelera- 
tion-transformation "  means  that  the  equations 
giving  the  transformation  from  the  variables  x,  y, 
z,  t  to  the  system  of  variables  xlt  x2,  x3,  #4,  which  is 
the  basis  of  our  discussion,  can  be  regarded  as  giving 
the  relations  between  two  systems  of  reference 
which  are  moving  with  an  accelerated  motion  rela- 
tively to  one  another.  The  nature  of  the  state  of 
motion  of  two  systems  of  reference  relative  to  one 
another  finds  its  expression  in  the  analytical  form 
of  the  equations  of  transformation  of  their  co- 
ordinates. 


THEORY  OF  GRAVITATION  99 

Note  26  (p.  51).  Two  things  are  to  be  under- 
taken in  the  following  :  (i)  the  fundamental  equa- 
tions of  the  new  theory  are  to  be  written  in  an 
explicit  form,  and  (2)  the  transition  to  Newton's 
fundamental  equations  is  to  be  performed. 

i.  From  the  equation  of  variation  8{S^s}  =  o 
where 

4 

ds2  =  Zg^dXpdXv, 
i 

we  have,  after  carrying  out  the  operation  of  varia- 
tion, the  four  total  differential  equations  : 


These  are  the  equations  of  motion  of  a  material  point 
in  the  gravitational  field  defined  by  the  g^'s. 
The  symbol  Y^v  here  denotes  the  expression 

W   ,   Sfr.        S 


The  symbol  gffa  denotes  the  minor  of  gu  in  the 
determinant 

ten,  ...,...,  £14] 


divided  by  the  determinant  itself. 

The  ten  differential  equations  for  the  "  gravita- 
tional potentials  "  g^  are  : 


The  quantities  T^  and  T  are  expressions  which 
are  related  in  a  simple  manner  to  the  components 
of  the  stress-energy-tensor  (which  plays  the  part 


100      THE  FOUNDATIONS  OF  EINSTEIN'S 

of  the  quantity  exciting  the  field  in  the  new  theory 
in  place  of  the  density  of  mass).  K  is  essentially 
equal  to  the  gravitational  constant  of  Newton's 
theory. 

The  differential  equations  (i)  and  (2)  are  the  funda- 
mental equations  of  the  new  theory.  The  derivation 
of  these  equations  is  carried  out  in  detail  in  the 
tract  by  A.  Einstein,  "  The  Foundations  of  the 
General  Principle  of  Relativity,"  J.  A.  Barth, 
Leipzig,  1916. 

2.  In  order  to  obtain  a  connection  between  these 
equations  and  Newton's  theory,  we  must  make 
several  simplifying  assumptions.  We  shall  first 
assume  that  the  g^'s  differ  only  by  quantities  which 
are  small  compared  with  unity  from  the  values 
given  by  the  scheme  : 

l     #12     #13     £l4" 

1  £22  £23  £24 
£31  £32  £33  £34 
^£41  £42  £43  £44- 

These  values  for  the  g^'s  characterize  the  case  of 
the  special  theory  of  relativity,  i.e.  the  case  of  the 
condition  free  of  gravitation.  We  shall  also  assume 
that,  at  infinite  distances,  the  gMI/s  tend  to,  and 
do  finally,  assume  the  above  values  ;  that  is,  that 
matter  does  not  extend  into  infinite  space. 

Secondly,  we  shall  assume  that  the  velocities  of 
matter  are  small  compared  with  the  velocity  of 
light,  and  can  be  regarded  as  small  quantities  of 
the  first  order.  The  quantities 

dxlt     dx2    dx$ 
~Js~'    ~ds'    ~fc' 

will  then  be  infinitely  small  quantities  of  the  first 

order,  and  -^i  will  equal  i,  except  for  quantities  of 
as 


THEORY  OF  GRAVITATION  101 

the  second  order.  From  the  equations  defining 
the  T*v  it  will  then  be  seen  that  these  quantities  will 
be  infinitely  small,  of  the  first  order.  If  we  neglect 
quantities  of  the  second  order,  and  finally  assume 
that,  for  small  velocities  of  matter,  the  changes  of 
the  gravitational  field  with  respect  to  time  are  small 
(i.e.  that  the  derivatives  of  the  gMI/s  with  respect 
to  time  may  be  neglected  in  comparison  with  the 
derivatives  taken  with  regard  to  the  space-co- 
ordinates) the  system  of  equations  (i)  assumes  the 
form  : 


This  would  be  the  equation  of  motion  of  a  point- 
mass  as  already  given  by  Newton's  mechanics,  if 
i#44  be  taken  as  representing  the  ordinary  gravita- 
tional potential.  It  still  remains  to  be  seen  what  the 
differential  equation  for  g44  becomes  in  the  new  theory 
under  the  simplifying  assumptions  we  have  chosen. 
The  stress-energy  tensor,  which  excites  the  field, 
degenerates,  as  a  result  of  our  quite  special  assump- 
tions, into  the  density  of  mass  p  : 


In  the  differential  equations  (2)  the  second  term 
on  the  left-hand  side  is  the  product  of  two  magni- 
tudes, which,  according  to  the  above  arguments, 
are  to  be  regarded  as  infinitely  small  quantities  of 
the  first  order.  Thus  the  second  term,  being  of 
the  second  order  of  small  quantities,  may  be  dis- 
missed. The  first  term,  on  the  other  hand,  if  we 
omit  the  terms  differentiated  with  respect  to  time, 
as  above  (i.e.  if  we  regard  the  gravitational  field 
as  "  stationary  "),  reduces  to  : 


=  v  =  4. 


102      THE  FOUNDATIONS  OF  EINSTEIN'S 

The  differential  equation  for  g44  thus  degenerates 
into  Poisson's  equation  : 

(2,0)  A#44  =  Kp. 

Thus,  to  a  first  approximation  (i.e.  if  one  regards  the 
velocity  of  light  as  infinitely  great,  and  this  is  a 
characteristic  feature  of  the  classical  theory,  as 
was  explained  in  detail  in  §  3  (b)  :  if  certain  simple 
assumptions  are  made  about  the  behaviour  of  the 
gMI/s  at  infinity  ;  and  if  the  time-changes  of  the 
gravitational  field  are  neglected)  the  well-known 
equations  of  Newtonian  mechanics  emerge  out  of 
the  differential  equations  of  Einstein's  theory, 
which  were  obtained  from  perfectly  general  begin- 
nings. 

Note  27  (p.  53).  The  theory  of  surfaces,  i.e. 
the  study  of  geometry  upon  surfaces,  makes  it 
immediately  apparent  that  the  theorems,  which 
have  been  established  for  any  surface,  also  hold 
for  any  surface  which  can  be  generated  by  dis- 
torting the  first  without  tearing.  For  if  two  surfaces 
have  a  point-to-point  correspondence,  such  that 
the  line-elements  are  equal  at  corresponding  points, 
then  corresponding  finite  arcs,  angles,  and  areas, 
etc.,  will  be  equal.  One  thus  arrives  at  the  same 
planimetrical  theorems  for  the  two  surfaces.  Such 
surfaces  are  called  "  deformable  "  surfaces.  The 
necessary  and  sufficient  condition  that  surfaces  be 
continuously  deformable  is  that  the  expression  for 
the  line-element  of  the  one  surface 


ds2  =  gudxS  +  g^Xjdxi  + 
can  be  transformed  into  that  for  the  other, 


According  to  Gauss,  it  is  necessary  that  both  sur- 
faces have  equal  measures  of  curvature.     If  the  latter 


THEORY  OF  GRAVITATION  103 

is  constant  over  the  whole  surface,  as  e.g.  in  the 
case  of  a  cylinder  or  a  plane,  all  conditions  for  the 
deformability  of  the  surfaces  are  fulfilled.  In  other 
cases,  special  equations  offer  a  criterion  as  to  whether 
surfaces,  or  portions  of  surfaces,  are  deformable 
into  one  another.  The  numerous  subsidiary  prob- 
lems, which  result  out  of  these  questions,  are 
discussed  at  length  in  every  book  dealing  with 
differential  geometry  (e.g.  Bianchi-Lukat).*  This 
branch  of  training,  which  was  hitherto  of  interest 
only  to  mathematicians,  now  assumes  very  con- 
siderable importance  for  the  physicist  too. 

Note  28  (p.  61).  One  must  avoid  being  de- 
ceived into  the  belief  that  Newton's  fundamental 
law  is  in  any  way  to  be  regarded  as  an  explanation 
of  gravitation.  The  conception  of  attractive  force 
is  borrowed  from  our  muscular  sensations,  and  has 
therefore  no  meaning  when  applied  to  dead  matter. 
C.  Neumann,  who  took  great  pains  to  place  Newton's 
mechanics  on  a  solid  basis,  glosses  upon  this  point 
himself  in  a  drastic  fashion,  in  the  following  narra- 
tive, which  shows  up  the  weaknesses  of  the  former 
view : 

"  Let  us  suppose  an  explorer  to  narrate  to  us  his 
experiences  in  yonder  mysterious  ocean.  He  had 
succeeded  in  gaining  access  to  it,  and  a  remarkable 
sight  had  greeted  his  eyes.  In  the  middle  of  the 
sea  he  had  observed  two  floating  icebergs,  a  larger 
and  a  smaller  one,  at  a  considerable  distance  from 
one  another.  Out  of  the  interior  of  the  larger  one, 
a  voice  had  resounded,  issuing  the  following  com- 
mand in  a  peremptory  tone  :  '  Ten  feet  nearer  !  ' 
The  little  iceberg  had  immediately  carried  out  the 
order,  approaching  ten  feet  nearer  the  larger 
one.  Again,  the  larger  gave  out  the  order  :  '  Six 
feet  nearer  !  '  The  other  had  again  immediately 

*  Forsyth's  "  Differential  Geometry."— H.  L.  B. 


104       THE  FOUNDATIONS  OF  EINSTEIN'S 

i 

executed  it.  And  in  this  manner  order  after  order 
had  echoed  out :  and  the  little  iceberg  had  con- 
tinually been  in  motion,  eager  to  put  every  command 
immediately  and  implicitly  into  action. 

"  We  should  certainly  consign  such  a  report  to 
the  realm  of  fables.  But  let  us  not  scoff  too  soon  ! 
The  ideas,  which  appear  so  extraordinary  to  us  in 
this  case,  are  exactly  the  same  as  those  which  lie 
at  the  base  of  the  most  complete  branch  of  natural 
science,  and  to  which  the  most  famous  of  physicists 
owes  the  glory  attached  to  his  name. 

"  For  in  cosmic  space  such  commands  are  con- 
tinually resounding,  proceeding  from  each  of  the 
heavenly  bodies — from  the  sun,  planets,  moons,  and 
comets.  Every  single  body  in  space  hearkens  to  the 
orders  which  the  other  bodies  give  it,  always  striving 
to  carry  them  out  punctiliously.  Our  earth  would 
dash  through  space  in  a  straight  line,  if  she  were 
not  controlled  and  guided  by  the  voice  of  command, 
issuing  from  moment  to  moment,  from  the  sun,  in 
which  the  instructions  of  the  remaining  cosmic 
bodies  are  less  audibly  mingled. 

"  These  commands  are  certainly  given  just  as 
silently  as  they  are  obeyed ;  and  Newton  has 
denominated  this  play  of  interchange  between  com- 
manding and  obeying  by  another  name.  He  talks 
quite  briefly  of  a  mutual  attractive  force,  which 
exists  between  cosmic  bodies.  But  the  fact  remains 
the  same.  For  this  mutual  influence  consists  in 
one  body  dealing  out  orders,  and  the  other  obeying 
them." 


THEORY  OF  GRAVITATION  105 


ON  THE  THEORY  OF  RELATIVITY 
By  Henry  L.  Brose,  M.A. 
INTRODUCTORY 

PHYSICS,  being  a  science  of  observation 
which  seeks  to  arrange  natural  phenomena 
into  a  consistent  scheme  by  using  the 
methods  and  language  of  mathematics,  has  to 
inquire  whether  the  assumptions  implied  in  any 
branch  of  mathematics  used  for  this  purpose  are 
legitimate  in  its  sphere,  or  whether  they  are  merely 
the  outcome  of  convention,  or  have  been  built  up 
from  abstract  notions  containing  foreign  elements. 
The  use  of  a  unit  length  as  an  unalterable  measure, 
or  of  a  time-division,  has  been  accepted  in  tradi- 
tional mechanics  without  inconsistency  manifesting 
itself  in  general  until  the  field  of  electrodynamics 
became  accessible  to  investigators  and  rendered  a 
re-examination  of  the  foundations  of  our  modes  of 
measurement  necessary.  It  is  upon  these  that  the 
whole  science  of  mathematical  physics  rests.  The 
road  of  advance  of  all  science  is  in  like  manner 
conditioned  by  the  inter-play  of  observations  and 
notions,  each  assisting  the  other  in  giving  us  a 
clearer  view  of  Nature  regarded  purely  as  a  physical 
reality.  The  discovery  of  additional  phenomena 
presages  a  still  greater  unification,  revealing  new 
relations  and  exposing  new  differences  ;  the  ultimate 
aim  of  physics  would  seem  to  consist  in  reaching 


106      THE  FOUNDATIONS  OF  EINSTEIN'S 

perfect  separation  and  distinctness  of  detail  simul- 
taneously with  perfect  co-ordination  of  the  whole. 
"  The  all-embracing  harmony  of  the  world  is  the 
true  source  of  beauty  and  is  the  real  truth,"  as 
Poincare  has  expressed  it.  The  noblest  task  of 
co-ordinating  all  knowledge  falls  to  the  lot  of  phil- 
osophy. 

A  principle  which  has  proved  fruitful  in  one 
sphere  of  physics  suggests  that  its  range  may  be 
extended  into  others  ;  nowhere  has  this  led  to  more 
successful  results  than  in  the  increasing  generaliza- 
tion which  has  characterized  the  advance  of  the 
principle  of  relativity.  This  advance  is  marked  by 
three  stages,  quite  distinct,  indeed,  in  the  nucleus 
of  their  growth,  yet  each  succeeding  stage  including 
the  results  of  the  earlier. 

Relativity  first  makes  its  appearance  as  a  govern- 
ing principle  in  Newtonian  or  Galilean  mechanics  ; 
difficulties  arising  out  of  the  study  of  the  phenomena 
of  radiations  led  to  a  new  enunciation  of  the  principle 
upon  another  basis  by  Einstein  in  1905,  an  enuncia- 
tion which  comprised  the  phenomena  of  both  me- 
chanics and  radiation:  this  will  be  referred  to  as 
the  "  special "  principle  of  relativity  to  distinguish  it 
from  the  "  general"  principle  of  relativity  enunciated 
by  Einstein  in  1915,  and  which  applied  to  all  physical 
phenomena  and  every  kind  of  motion.  The  latter 
theory  also  led  to  a  new  theory  of  gravitation. 


I.  THE  MECHANICAL  THEORY  OF 
RELATIVITY 

In  order  to  arrive  at  the  precise  significance  of 
the  principle  of  relativity  in  the  form  in  which  it 
held  sway  in  classical  mechanics,  we  must  briefly 
analyse  the  terms  which  will  be  used  to  express  it. 


THEORY  OF  GRAVITATION  107 

Mechanics  is  usually  denned  as  the  science  which 
describes  how  the  "  position  of  bodies  in  '  space  ' 
alters  with  the  '  time.'  '  We  shall  for  the  present 
discuss  only  the  term  "  position,"  which  also  in- 
volves "  distance/'  leaving  time  and  space  to  be 
dealt  with  later  when  we  have  to  consider  the  mean- 
ing of  physical  simultaneity.  Modern  pure  geometry 
starts  out  from  certain  conceptions  such  as  "  point," 
"  straight  line,"  and  "  plane,"  which  were  originally 
abstracted  from  natural  objects  and  which  are 
implicitly  denned  by  a  number  of  irreducible  and 
independent  axioms  ;  from  these  a  series  of  pro- 
positions is  deduced  by  the  application  of  logical 
rules  which  we  feel  compelled  to  regard  as  legitimate. 
The  great  similarity  which  exists  between  geometri- 
cally constructed  figures  and  objects  in  Nature  has 
led  people  erroneously  to  regard  these  propositions 
as  true  :  but  the  truth  of  the  propositions  depends 
on  the  truth  of  the  axioms  from  which  the  proposi- 
tions were  logically  derived.  Now  empirical  truth 
implies  exact  correspondence  with  reality.  But 
pure  geometry  by  the  very  nature  of  its  genesis 
excludes  the  test  of  truth.  There  are  no  geometrical 
points  or  straight  lines  in  Nature,  nor  geometrical 
surfaces  ;  we  only  find  coarse  approximations  which 
are  helpful  in  representing  these  mathematically 
conceived  elements. 

If,  however,  certain  principles  of  mechanics  are 
conjoined  with  the  axioms  of  geometry,  we  leave 
the  realm  of  pure  geometry  and  obtain  a  set  of 
propositions  which  may  be  verified  by  comparison 
with  experience,  but  only  within  limits,  viz.  in 
respect  to  numerical  relations,  for  again  no  exact 
correspondence  is  possible,  merely  a  superposition  of 
geometrical  points  with  places  occupied  by  matter. 
Our  idea  of  the  form  of  space  is  derived  from  the 
behaviour  of  matter,  which,  indeed,  conditions  it. 


108    THE  FOUNDATIONS  OF  EINSTEIN'S 

Space  itself  is  amorphous,  and  we  are  at  liberty  to 
build  up  any  geometry  we  choose  for  the  purpose 
of  making  empirical  content  fit  into  it.  Neither 
Euclidean,  nor  any  of  the  forms  of  meta-geometry, 
has  any  claim  to  precedence.  We  may  select  for 
a  consistent  description  of  physical  phenomena 
whichever  is  the  more  convenient,  and  requires  a 
minimum  of  auxiliary  hypotheses  to  express  the 
laws  of  physical  nature. 

Applied  geometry  is  thus  to  be  treated  as  a  branch 
of  physics.  We  are  accustomed  to  associate  two 
points  on  a  straight  line  with  two  marks  on  a 
(practically)  rigid  body  :  when  once  we  have  chosen 
an  arbitrary,  rigid  body  of  reference,  we  can  discuss 
motions  or  events  mechanically  by  using  the  body 
as  the  seat  of  a  set  of  axes  of  co-ordinates.  The 
use  of  the  rule  and  compasses  gives  us  a  physical 
interpretation  of  the  distance  between  points,  and 
enables  us  to  state  this  distance  by  measurement 
numerically,  inasmuch  as  we  may  fix  upon  an 
arbitrary  unit  of  length  and  count  how  often  it  has 
to  be  applied  end  to  end  to  occupy  the  distance 
between  the  points.  Every  description  in  space 
of  the  scene  of  an  event  or  of  the  position  of  a  body 
consists  in  designating  a  point  or  points  on  a  rigid 
body  imagined  for  the  purpose,  which  coincides  with 
the  spot  at  which  the  event  takes  place  or  the  object 
is  situated.  We  ordinarily  choose  as  our  rigid 
body  a  portion  of  the  earth  or  a  set  of  axes  attached 
to  it. 

Now  Newton's  (or  Galilei's)  law  of  motion  states 
that  a  body  which  is  sufficiently  far  removed  from 
all  other  bodies  continues  in  its  state  of  rest  or 
uniform  motion  in  a  straight  line.  This  holds  very 
approximately  for  the  fixed  stars.  If,  however,  we 
refer  the  motion  of  the  stars  to  a  set  of  axes  fixed 
to  the  earth,  the  stars  describe  circles  of  immense 


THEORY  OF  GRAVITATION  109 

radius ;  that  is,  for  such  a  system  of  reference  the 
law  of  inertia  only  holds  approximately.  Hence 
we  are  led  to  the  definition  of  Galilean  systems  of 
co-ordinates.  A  Galilean  system  is  one,  the 
state  of  motion  of  which  is  such  that  the  law 
of  inertia  holds  for  it.  It  follows  naturally  that 
Newtonian  or  Galilean  mechanics  is  valid  only  for 
such  Galilean  or  inertial  systems  of  co-ordinates,  i.e. 
in  formulating  expressions  for  the  motion  of  bodies 
we  must  choose  some  such  system  at  an  immense 
distance  where  the  Newtonian  law  would  hold.  It 
will  be  noticed  that  this  is  an  abstraction,  and  that 
such  a  system  is  merely  postulated  by  the  law  of 
motion.  It  is  the  foundation  of  classical  mechanics, 
and  hence  also  of  the  first  or  "  mechanical  "  principle 
of  relativity. 

If  we  suppose  a  crow  flying  in  a  straight  line 
at  uniform  velocity  with  respect  to  the  earth  dia- 
gonally over  a  train  likewise  moving  uniformly  and 
rectilinearly  with  respect  to  the  earth  (since  motion 
is  change  of  position  we  must  specify  our  rigid  body 
of  reference,  viz.  the  earth),  then  an  observer  in 
the  train  would  also  see  the  crow  flying  in  a  straight 
line,  but  with  a  different  uniform  velocity,  judged 
from  a  system  of  co-ordinates  attached  to  the  train. 
We  may  consider  both  the  train  and  the  earth  to  be 
carriers  of  inertial  systems  as  we  are  only  dealing 
with  small  distances.  We  can  then  formulate  the 
mechanical  principle  of  relativity  as  follows  : — 

If  a  body  be  moving  uniformly  and  rectilinearly 
with  respect  to  a  co-ordinate  system  K  then  it  will 
likewise  move  uniformly  and  rectilinearly  with  re- 
spect to  a  second  co-ordinate  system  K',  provided 
that  the  latter  be  moving  uniformly  and  rectilinearly 
with  respect  to  the  first  system  K. 

In  our  illustration,  the  crow  represents  the  body, 
K  is  the  earth,  and  K'  is  carried  by  the  train. 


110      THE  FOUNDATIONS  OF  EINSTEIN'S 

Or,  we  may  say  that  if  K  be  an  inertial  system 
then  K',  which  moves  uniformly  and  rectilinearly 
with  respect  to  K,  is  also  an  inertial  system.  Hence, 
since  the  laws  of  Newtonian  mechanics  are  based 
on  inertial  systems,  it  follows  that  all  such  systems 
are  equivalent  for  the  description  of  the  laws  of 
mechanics  :  no  one  system  amongst  them  is  unique, 
and  we  cannot  define  absolute  motion  or  rest ;  any 
systems  moving  with  mutual  rectilinear  uniform 
motion  may  be  regarded  as  being  at  rest.  Mathe- 
matically, this  means  that  the  laws  of  mechanics 
remain  unchanged  in  form  for  any  transformation 
from  one  set  of  inertial  axes  to  another. 

The  development  of  electrodynamics  and  the 
phenomena  of  radiation  generally  showed,  however, 
that  the  laws  of  radiation  in  one  inertial  system  did 
not  preserve  their  form  when  referred  to  another 
inertial  system  :  K  and  K'  were  no  longer  equivalent 
for  the  description  of  phenomena  such  as  that  of 
light  passing  through  a  moving  medium.  This  meant 
that  either  there  was  a  unique  inertial  system 
enabling  us  to  define  absolute  motion  and  rest  in 
nature,  or  that  we  would  have  to  build  up  a  theory 
of  relativity,  not  on  the  inertial  law  and  inertial 
systems,  but  on  some  new  foundation  which  would 
definitely  ensure  that  the  form  of  all  physical  laws 
would  be  preserved  in  passing  from  one  system  of 
reference  to  another. 

This  dilemma  arose  out  of  the  conflicting  results 
of  two  experiments,  viz.  Fizeau's  and  Michelson 
and  Morley's. 

Fizeau's  experiment  was  designed  to  determine 
whether  the  velocity  of  light  through  moving 
liquid  media  was  different  from  that  through  a 
stationary  medium,  i.e.  whether  the  motion  of  the 
liquid  caused  a  drag  on  the  aether,  which  it  would 
do  if  the  mechanical  law  of  relativity  held  for  light 


THEORY  OF  GRAVITATION  111 

phenomena,  for  then  the  light  ray  would  be  in  the 
same  position  as  a  swimmer  travelling  upstream  or 
downstream  respectively.* 

No  "ether-drag"  was,  however,  detected;  only  a 
fraction  of  the  velocity  of  the  liquid  seemed  to  be 
added  to  the  velocity  of  light  (c)  under  ordinary 
conditions,  and  this  fraction  depends  on  the  refractive 
index  of  the  liquid,  and  had  previously  been  calcu- 
lated by  Fresnel :  for  a  vacuum  this  fraction 
vanishes. 

This  result  seemed  to  favour  the  hypothesis  of  a 
fixed  ether,  as  was  supported  by  Fresnel  and 
Lorentz.  But  a  fixed  ether  implies  that  we  should 
be  able  to  detect  absolute  motion,  that  is,  motion 
with  respect  to  the  ether. 

Arguing  from  this,  let  us  consider  an  observer  in 
the  liquid  moving  with  it.  //  there  is  a  fixed 
ether,  he  should  find  a  lesser  value  for  the  velocity 
of  light  (i.e.  <  c)  owing  to  his  own  velocity  in  the 
same  direction,  or  vice  versa  in  the  opposite 
direction. 

But  we  on  the  earth  are  in  the  position  of  the 
observer  in  the  liquid  since  we  revolve  around  the 
sun  at  the  rate  of,  approximately,  30  kms.  per  second 

(i.e.   — - — ),  and  we  are  subject  to   a  translatory 
io,ooo; 

motion  of  about  the  same  magnitude :  hence  we 
should  be  able  to  detect  a  change  in  the  velocity 
of  light  due  to  our  change  of  motion  through  the 
ether.  These  considerations  give  rise  to  Michelson 
and  Morley's  experiment. 

Michelson  and  Morley  attempted  to  detect  motion 
relative  to  the  supposedly  fixed  ether  by  the  inter- 
ference of  two  rays  of  light,  one  travelling  in  the 

*  It  is  well  known  that  it  takes  a  swimmer  longer  to  travel  a 
certain  distance  up  and  down  stream  than  to  swim  across  the  stream 
and  back  an  equal  distance. 


THE  FOUNDATIONS  OF  EINSTEIN'S 

direction  of  motion  of  the  earth's  velocity,  the  other 
travelling  across  this  direction  of  motion. 

No  change  in  the  initial  interference  bands  was, 
however,  observed  when  the  position  of  the  instru- 
ment was  changed,  although  such  an  effect  was 
easily  within  the  limits  of  accuracy  of  the  experi- 
ment. Many  modifications  of  the  experiment  like- 
wise failed  to  demonstrate  the  presence  of  an 
"  ether-wind." 

To  account  for  these  negative  results  as  con- 
tradicting deductions  from  Fizeau's  experiment, 
Fitzgerald  and,  later,  independently,  Lorentz  sug- 
gested the  theory  that  bodies  automatically  contract 
when  moving  through  the  ether,  and  since  our 
measuring  scales  contract  in  the  same  ratio,  we  are 
unable  to  detect  this  alteration  in  length  ;  this 
effect  would  lead  us  always  to  get  the  same  result 
for  the  velocity  of  light.  This  contraction-hypoth- 
esis agrees  well  with  the  electrical  theory  of  matter 
and  may  be  attributed  to  changes  in  the  electro- 
magnetic forces,  acting  between  particles,  which 
determine  the  equilibrium  of  a  so-called  rigid  body. 

Thus  Michelson  and  Morley's  experiment  seems  to 
prove  that  the  principle  of  relativity  of  mechanics 
also  holds  for  radiation  effects,  that  is,  it  is  impossible 
to  determine  absolute  motion  through  the  ether  or 
space  :  this  implies  that  there  is  no  unique  system 
of  co-ordinates.  It  disagrees  with  Fizeau's  result 
and  seems  to  indicate  the  existence  of  a  "  moving 
ether,"  i.e.  an  ether  which  is  carried  along  by 
moving  bodies,  as  was  upheld  by  Stokes  and  Hertz. 
Lord  Rayleigh  pointed  out  that  if  the  contraction- 
hypothesis  of  Lorentz  and  Fitzgerald  were  true, 
isotropic  bodies  ought  to  become  anisotropic  on 
account  of  the  motion  of  the  earth,  and  that  conse- 
quently, phenomena  of  double  refraction  should 
make  their  appearance.  Experiments  which  he 


THEORY  OF  GRAVITATION  113 

himself  conducted  with  carbon  bisulphide  and 
others  carried  out  by  Bruce  with  water  and  glass 
produced  a  negative  result. 

II.  THE  "  SPECIAL  "  THEORY  OF 
RELATIVITY 

Einstein,  in  the  special  theory  of  relativity, 
surmounts  these  difficulties  by  doing  away  with  the 
ether  (as  a  substance)  and  assumes  that  light- 
signals  project  themselves  as  such  through  space. 
Faraday  had  already  long  ago  expressed  the  opinion 
that  the  field  in  which  radiations  take  place  must 
not  be  founded  upon  considerations  of  matter,  but 
rather  that  matter  should  be  regarded  as  singu- 
larities or  places  of  a  singular  character  in  the 
field.  We  may  retain  the  name  "  ether  "  for  the 
field  as  long  as  we  do  not  regard  it  as  composed  of 
matter  of  the  kind  we  know.  Einstein  arrives  at 
these  conclusions  by  critically  examining  our  notions 
of  space  and  time  or  of  distance  and  simultaneity. 

We  know  what  simultaneity  (time-coincidence  of 
two  events)  means  for  our  consciousness,  but  in 
making  use  of  the  idea  of  simultaneity  in  physics, 
we  must  be  able  to  prove  by  actual  experiment 
or  observation  that  two  events  are  simultaneous 
according  to  some  definition  of  simultaneity.  A 
conception  only  has  meaning  for  the  physicist 
if  the  possibility  of  verifying  that  it  agrees 
with  actual  experience  is  given.  In  other  words, 
we  must  have  a  definition  of  simultaneity  which 
gives  us  an  immediate  means  of  proving  by  experi- 
ment whether,  e.g.  two  lightning-strokes  at  different 
places  occur  simultaneously  for  an  observer  situated 
somewher  ebetween  them  or  not.  Whenever  measure- 
ments are  undertaken  in  physics  two  points  are  made 
to  coincide,  whether  they  be  marks  on  a  scale  and  on 
an  object,  or  whether  they  be  cross-wires  in  a  tele- 
8 


114      THE  FOUNDATIONS  OF  EINSTEIN'S 

scope  which  have  been  made  to  coincide  with  a  distant 
object  to  allow  angular  measurements  to  be  made  ; 
coincidence  is  the  only  exact  mode  of  observation,  and 
lies  at  the  bottom  of  all  physical  measurements.  The 
same  importance  attaches  to  simultaneity,  which  is 
coincidence  in  time.  It  is  to  be  noted  that  no 
definition  will  be  made  for  simultaneity  occurring 
at  (practically)  one  point :  for  this  case  psychological 
simultaneity  must  be  accepted  as  the  basis  :  the 
necessity  for  a  physical  definition  arises  only  when 
two  events  happening  at  great  distances  apart  are 
to  be  compared  as  regards  the  moment  of  their 
happening.  We  cannot  do  more  than  reduce  the 
simultaneity  of  two  events  happening  a  great  dis- 
tance apart  to  simultaneity  referred  to  a  single 
observer  at  one  point :  this  would  satisfy  the  re- 
quirements of  physics. 

Einstein,  accepting  Michelson  and  Morley's  result, 
introduced  the  convention  in  1905  that  light  is 
propagated  with  a  constant  velocity  ( =  c,  i.e. 
300,000  kms.  per  sec.  approximately)  in  vacuo  in  all 
directions,  and  he  then  makes  use  of  light-signals 
to  connect  up  two  events  in  time. 

He  illustrates  his  line  of  argument  roughly  by 
assuming  two  points,  A  and  B,  very  far  apart  on  a 
railway  embankment  and  an  observer  at  M  midway 
between  A  and  B,  provided  with  a  contrivance 
such  as  two  mirrors  inclined  at  90°  and  adjusted  so 
that  light  from  A  and  B  would  be  reflected  into  his 
vertical  line  of  sight  (Fig.  3). 

Two  events  such  as  lightning-strokes  are  then  to 
be  defined  as  simultaneous  for  the  observer  at  M 
if  rays  of  light  from  them  reach  the  observer  at  the 
same  moment  (psychologically)  :  i.e.  if  he  sees  the 
strokes  in  his  mirror-contrivance  simultaneously. 

Next  suppose  that  a  very  long  train  is  moving 
with  very  great  uniform  velocity  along  the  embank- 


THEORY  OF  GRAVITATION  115 

ment,  and  that  the  lightning-strokes  pass  through  the 
two  corresponding  points  A1  and  B1  of  the  train  thus : 


Train 


MM//w/Mf/M 

/  Embankment  / 

FIG.  3. 

The  question  now  arises  :  Are  the  two  lightning- 
strokes  at  A  and  B,  which  are  simultaneous  with 
respect  to  the  embankment  also  simultaneous  with 
respect  to  the  moving  train  ?  It  is  quite  clear  that 
as  M1  is  moving  towards  B1  and  away  from  A1,  the 
observer  at  M1  (mid-point  of  A1B1)  will  receive  the 
ray  emitted  from  B1  sooner  than  that  emitted  from 
A1  and  he  would  say  that  the  lightning-stroke  at 
B  or  B1  occurred  earlier  than  the  one  at  A  or  A1. 
Hence  our  condition  of  simultaneity  is  not  satisfied 
and  we  are  forced  to  the  conclusion  that  events 
which  are  simultaneous  for  one  rigid  body  of  refer- 
ence (the  embankment)  are  not  simultaneous  for 
another  body  of  reference  (the  train)  which  is  in 
motion  with  regard  to  the  first  rigid  body  of  reference. 
This  establishes  the  relativity  of  simultaneity. 

This  is,  of  course,  only  an  elementary  example  of 
a  very  special  case  of  the  regulation  of  clocks  by 
light-signals.  It  may  be  asked  how  the  mid-point 
M  is  found  :  one  might  simply  fix  mirrors  at  A  and 
B  and  by  flashing  light-signals  from  points  between 
A  and  B  ascertain  by  trial  the  point  (M)  at  which 
the  return-flashes  are  observed  simultaneously  :  this 
makes  M  the  mid-point  between  the  "  time  "-distance 
from  A  and  B  on  the  embankment. 

The  relativity  of  simultaneity  states  that  every 
rigid  body  of  reference  (co-ordinate  system)  has  its 
own  time  :  a  time-datum  only  has  meaning  when  the 


116      THE  FOUNDATIONS  OF  EINSTEIN'S 

body  of  reference  is  specified,  or  we  may  say  that 
simultaneity  is  dependent  on  the  state  of  motion  of 
the  body  of  reference. 

Similar  reasoning  applies  in  the  case  of  the  distance 
between  two  points  on  a  rigid  body.  The  length 
of  a  rod  is  defined  as  the  distance,  measured 
by  (say)  a  metre-rule,  between  the  two  points 
which  are  occupied  simultaneously  by  the  two 
ends.  Since  simultaneity,  as  we  have  just  seen,  is 
relative,  the  distance  between  two  points,  since  they 
depend  on  a  simultaneous  reading  of  two  events,  is 
also  relative,  and  length  only  has  a  meaning  if  the 
body  of  reference  is  likewise  specified  :  any  change 
of  motion  entails  a  corresponding  change  of  length  : 
we  cannot  detect  the  change  since  our  measures 
alter  in  the  same  ratio.  Length  is  thus  a  relative 
conception,  and  only  reveals  a  relation  between 
the  observer  and  an  object:  the  "actual"  length 
of  a  body  in  the  sense  we  usually  understand  it 
does  not  exist :  there  is  no  meaning  in  the  term. 
The  length  of  a  body  measured  parallel  to  its  direc- 
tion of  motion  will  always  yield  a  greater  result 
when  judged  from  a  system  attached  to  it  than 
from  any  other  system.  These  few  remarks  may 
suffice  to  indicate  the  relativity  of  distance. 

In  classical  mechanics  it  had  always  been  assumed 
that  the  time  which  elapsed  between  the  happening 
of  two  events,  and  also  the  distance  between  two 
points  of  a  rigid  body  were  independent  of  the  state 
of  motion  of  the  body  of  reference  :  these  hypoth- 
eses must,  as  a  result  of  the  relativity  of  simul- 
taneity and  distance,  be  rejected.  We  may  now  ask 
whether  a  mathematical  relation  between  the  place 
and  time  of  occurrence  of  various  events  is  possible, 
such  that  every  ray  of  light  travels  with  the  same 
constant  velocity  c  whichever  rigid  body  of  reference 
be  chosen,  e.g.  such  that  the  rays  measured  by  an 


THEORY  OF  GRAVITATION  117 

observer  either  in  the  train  or  on  the  embankment 
travel  with  the  same  apparent  velocity. 

In  other  words,  if  we  assume  the  constancy  of 
propagation  of  light  in  vacuo  for  two  systems,  K 
and  K1  moving  uniformly  and  rectilinearly  with 
respect  to  one  another,  what  are  the  values  of  the 
co-ordinates  x1,  y1,  z1,  tl  of  an  event  with  respect  to 
K1,  if  the  values  x,  y,  z,  t  of  the  same  event  with 
respect  to  K  are  given  ? 

It  is  easy  to  arrive  at  this  so-called  Lorentz- 
Einstein  transformation,  e.g.  in  the  case  where  K1  is 
moving  relative  to  K  parallel  to  K's  x  axis  with 
uniform  velocity  v  we  get : 

x  —  vt  ,  , 

X1  = ______      y1  =  y      z1  =  Z. 


If  we  put  x  —  ct,  we  find  that  —  reduces  to  c. 

i.e.    c  =  -  =  ^-  is  the  same  for  both  systems,  and 

the  condition  of  the  constancy  of  c,  the  velocity 
of  light  in  vacuo,  is  preserved. 

If  N/I  —  v 2/c2  is  to  be  real,  then  v  cannot  be 
greater  than  c,  i.e.  c  is  the  limiting  or  maximum 
velocity  in  nature  and  has  thus  a  universal  signifi- 
cance. 

If  we  imagine  c  to  be  infinitely  great  in  com- 
parison with  v  (and  this  will  be  the  case  for  all 
ordinary  velocities,  such  as  those  which  occur  in 
mechanics),  the  equations  of  transformation  de- 
generate into  : 

x1  =  x  —  vt    y1  =  y     z1  =  z     t1  =  t. 

This  is  the  familiar  Galilean  transformation  which 
holds  for  the  "  mechanical  "  principle  of  relativity. 


118        THE  FOUNDATIONS  OF  EINSTEIN'S 

We  see  that  the  Lorentz-Einstein  transformation 
covers  both  mechanical  and  radiational  phenomena. 
The  special  theory  of  relativity  may  now  be 
enunciated  as  follows  :  All  systems  of  reference 
which  are  in  uniform  rectilinear  motion  with 
regard  to  one  another  can  be  used  for  the 
description  of  physical  events  with  equal  justi- 
fication. That  is,  if  physical  laws  assume  a 
particularly  simple  form  when  referred  to  any 
particular  system  of  reference,  they  will  preserve 
this  form  when  they  are  transformed  to  any  other 
co-ordinate  system  which  is  in  uniform  rectilinear 
motion  relatively  to  the  first  system.  The 
mathematical  significance  of  the  Lorentz-Einstein 
equations  of  transformation  is  that  the  expression 
for  the  infinitesimal  length  of  arc  ds 

(viz.  ds*  =  dx*  +  dy*  +  dz*  -  c*  dt2) 

in  the  space-time  *  manifold  x,  y,  z,  t,  preserves  its 
form  for  all  systems  moving  uniformly  and  rec- 
tilinearly  with  respect  to  one  another. 

Interpreted  geometrically  this  means  that  the 
transformation  is  conformal  in  imaginary  space  of 
four  dimensions.  Moreover,  the  time-co-ordinate 
enters  into  physical  laws  in  exactly  the  same  way 
as  the  three  space-co-ordinates,  i.e.  we  may  regard 
time  spatially  as  a  fourth  dimension  of  space.  This 
has  been  very  beautifully  worked  out  by  Minkowski, 
whose  premature  loss  is  deeply  to  be  regretted.  It 
may  be  fitting  here  to  recall  some  remarks  of  Berg- 
son  in  his  "  Time  and  Free  Will."  He  there  states 
that  "  time  is  the  medium  in  which  conscious  states 
form  discrete  series  :  this  time  is  nothing  but  space, 
and  pure  duration  is  something  different."  Again, 

*  A  continuous  manifold  may  be  defined  as  any  continuum  of 
elements  such  that  a  single  element  is  defined  by  n  continuously 
variable  magnitudes. 


THEORY  OF  GRAVITATION  119 

"  what  we  call  measuring  time  is  nothing  but  counting 
simultaneities ;  owing  to  the  fact  that  our  con- 
sciousness has  organized  the  oscillations  of  a  pendu- 
lum as  a  whole  in  memory,  they  are  first  preserved 
and  afterwards  disposed  in  a  series  :  in  a  word,  we 
create  for  them  a  fourth  dimension  of  space,  which 
we  call  homogeneous  time,  and  which  enables  the 
movement  of  the  pendulum,  although  taking  place 
at  one  spot,  to  be  continually  set  in  juxta-posi- 
tion  to  itself.  Duration  thus  assumes  the  illusory 
form  of  a  homogeneous  medium  and  the  connecting 
link  between  these  two  terms,  space  and  duration, 
is  simultaneity,  which  might  be  defined  as  the 
intersection  of  time  and  space."  Minkowski  calls 
the  space-time-manifold  "  world  "  and  each  point 
(event)  "  world-point." 

The  results  achieved  by  the  special  theory  of 
relativity  may  be  tabulated  as  follows  : — 

(1)  It  gives  a  consistent  explanation  of  Fizeau's 

and  Michelson  and  Morley's  experiment. 

(2)  It  leads  mathematically  at  once  to  the  value 

suggested  by  Fresnel  and  experimentally 
verified  by  Fizeau  for  the  velocity  of  a  beam 
of  light  through  a  moving  refracting  medium 
without  making  any  hypothesis  about  the 
physical  nature  of  the  liquid. 

(3)  It  gives  the  contraction  in  the  direction  of 

motion  for  electrons  moving  with  high  speed, 
without  requiring  any  artificial  hypothesis 
such  as  that  of  Lorentz  and  Fitzgerald  to 
explain  it. 

(4)  It  satisfactorily  explains  aberration,  i.e.  the 

influence  of  the  relative  motion  of  the  earth 
to  the  fixed  stars  upon  the  direction  of  motion 
of  the  light  which  reaches  us. 

(5)  It  accounts  for  the  influence  of  the  radial 

component  of  the  motion  of  the  stars,    as 


120     THE  FOUNDATIONS  OF  EINSTEIN'S 

shown  by  a  slight  displacement  of  the  spectral 
lines  of  the  light  which  reaches  us  from  the 
stars  when  compared  with  the  position  of  the 
same  lines  as  produced  by  an  earth  source. 

(6)  It  accounts  for  the  "fine  structure"  of  the 

spectral  lines  emitted  by  the  atom.* 

(7)  It   gives   the   expression   for  the  increase   of 

inertia,  owing  to  the  addition  of  (apparent) 
electromagnetic  inertia  of  a  charged  body  in 
motion. 

The  last  result,  however,  introduces  an  anomaly 
inasmuch  as  the  inertial  mass  of  a  quickly-moving 
body  increases,  but  not  the  gravitational  mass,  i.e. 
there  is  an  increase  of  inertia  without  a  corresponding 
increase  of  weight  asserting  itself.  One  of  the  most 
firmly  established  facts  in  all  physics  is  hereby 
transgressed.  This  result  of  the  theory  suggested 
a  new  basis  for  a  more  general  theory  of  relativity, 
viz.  that  proposed  by  Einstein  in  1915.  As  the 
special  theory  of  relativity  deals  only  with  uniform, 
rectilinear  motions,  its  structure  is  not  affected  by 
any  alteration  of  the  ideas  underlying  gravitation. 

III.  THE  GENERAL  THEORY  OF  RELATIVITY 

We  have  seen  that  the  first  or  "  mechanical  " 
theory  of  relativity  was  built  up  on  the  notion  of 
inertial  systems  as  deduced  from  the  law  of  inertia ; 
the  "  special  "  theory  of  relativity  was  built  up  on 
the  universal  significance  and  invariance  of  c,  the 
velocity  of  light  in  vacuo  ;  the  third  or  general  form 
of  relativity  is  to  be  founded  on  the  principle  of  the 
equality  of  inertial  and  gravitational  mass  and  in 
contradistinction  to  the  other  two  is  to  hold  not  only 
for  systems  moving  uniformly  and  rectilinearly  with 
respect  to  one  another,  but  for  all  systems  whatever 

*  See  Sommerfeld,  "  Atomic  Structure  and  Spectral  Lines,"  p.  474. 


THEORY  OF  GRAVITATION  121 

their  motion  ;  i.e.  physical  laws  are  to  preserve 
their  form  for  any  arbitrary  transformation  of  the 
variables  from  one  system  to  another. 

Mass  enters  into  the  formulae  of  the  older  physics 
in  two  forms  :  (i)  Force  =  inertial  mass  multiplied 
by  the  acceleration.  (2)  Force  —  gravitational  mass 
multiplied  by  the  intensity  of  the  field  of  gravitation  ; 
or, 

p  =  m  .  a    p  =  ml.g 

m1 

«~w* 

Observation  tells  us  that  for  a  given  field  of  gravi- 
tation the  acceleration  is  independent  of  the  nature 
and  state  of  a  body  ;  this  means  that  the  propor- 
tionality between  the  two  characteristic  masses 
(inertial  and  gravitational)  must  be  the  same  for  all 
bodies.  By  a  suitable  choice  of  units  we  can  make 
the  factor  of  proportionality  unity,  i.e.  m  =  m1. 

This  fact  had  been  noticed  in  classical  mechanics, 
but  not  interpreted. 

Eotvos  in  1891  devised  an  experiment  to  test  the 
law  of  the  equality  of  inertial  and  gravitational  mass  : 
he  argued  that  if  the  centre  of  inertia  of  a  hetero- 
geneous body  did  not  coincide  with  the  centre  of 
gravity  of  the  same  body,  the  centrifugal  forces  act- 
ing on  the  body  due  to  the  earth's  rotation  acting  at 
the  centre  of  inertia  would  not,  when  combined  with 
the  gravitational  forces  acting  at  the  centre  of  gravi- 
tational mass,  resolve  into  a  single  resultant,  but  that 
a  torque  or  turning  couple  would  exist  which  would 
manifest  itself,  if  the  body  were  suspended  by  a  very 
delicate  torsionless  thread  or  filament.  His  experi- 
ment disclosed  that  the  law  of  proportionality  of 
inertial  and  gravitational  mass  is  obeyed  with 
extreme  accuracy  :  fluctuations  in  the  ratio  could 
only  be  less  than  a  twenty-millionth. 


THE  FOUNDATIONS  OF  EINSTEIN'S 

Einstein  hence  assumes  the  exact  validity  of  the 
law,  and  asserts  that  inertia  and  gravitation  are 
merely  manifestations  of  the  same  quality  of  a 
body  according  to  circumstances.  As  an  illustra- 
tion of  the  purport  of  this  equivalence  he  takes  the 
case  of  an  observer  enclosed  in  a  box  in  free  space 
(i.e.  gravitation  is  absent)  to  the  top  of  which  a 
hook  is  fastened.  Some  agency  or  other  pulls  this 
hook  (and  together  with  it  the  box)  with  a  constant 
force.  To  an  observer  outside,  not  being  pulled,  the 
box  will  appear  to  move  with  constant  acceleration 
upwards,  and  finally  acquire  an  enormous  velocity. 
But  how  would  the  observer  in  the  box  interpret  the 
state  of  affairs  ?  He  would  have  to  use  his  legs  to 
support  himself  and  this  would  give  him  the  sensa- 
tion of  weight.  Objects  which  he  is  holding  in  his 
hands  and  releases  will  fall  relatively  to  the  floor 
with  acceleration,  for  the  acceleration  of  the  box  will 
no  longer  be  communicated  to  them  by  the  hand  ; 
moreover,  all  bodies  will  "  fall  "  to  the  floor  with  the 
same  acceleration.  The  observer  in  the  box,  whom 
we  suppose  to  be  familiar  with  gravitational  fields, 
will  conclude  that  he  is  situated  in  a  uniform  field  of 
gravitation  :  the  hook  in  the  ceiling  will  lead  him  to 
suppose  that  the  box  is  suspended  at  rest  in  the 
field  and  will  account  for  the  box  not  falling  in  the 
field.  Now  the  interpretation  of  the  observer  in  the 
box  and  the  observer  outside,  who  is  not  being  pulled, 
are  equally  justifiable  and  valid,  as  long  as  the 
equality  of  inertial  and  gravitational  mass  is  main- 
tained. 

We  may  now  enunciate  Einstein's  Principle  of 
Equivalence :  Any  change  which  an  observer  per- 
ceives in  the  passage  of  an  event  to  be  due  to  a 
gravitational  field  would  be  perceived  by  him  ex- 
actly in  the  same  way,  if  the  gravitational  field 
were  not  present  and  provided  that  he — the  ob- 


THEORY  OF  GRAVITATION 

server — make  his  system  of  reference  move  with 
the  acceleration  which  was  characteristic  of  the 
gravitation  at  his  point  of  observation. 

It  might  be  concluded  from  this  that  one  can 
always  choose  a  rigid  body  of  reference  such  that, 
with  respect  to  it,  no  gravitational  field  exists, 
i.e.  the  gravitational  field  may  be  eliminated  ;  this, 
however,  only  holds  for  particular  cases.  It  would 
be  impossible,  for  example,  to  choose  a  rigid  body 
of  reference  such  that  the  gravitational  field  of  the 
earth  with  respect  to  it  vanishes  entirely. 

The  principle  of  equivalence  enables  us  theoretic- 
ally to  deduce  the  influence  of  a  gravitational  field 
on  events,  the  laws  of  which  are  known  for  the 
special  case  in  which  the  gravitational  field  is  absent. 

We  are  familiar  with  space- time-domains,  which 
are  approximately  Galilean  when  referred  to  an 
appropriate  rigid  body  of  reference.  If  we  refer 
such  a  domain  to  a  rigid  body  of  reference  K1  moving 
irregularly  in  any  arbitrary  fashion,  we  may  assume 
that  a  gravitational  field  varying  both  with  respect 
to  time  and  to  space  is  present  for  K1 :  the  nature 
of  this  field  depends  on  the  choice  of  the  motion  of 
K1.  This  enabled  Einstein  to  discover  the  laws 
which  a  gravitational  field  itself  satisfies.  It  is  im-. 
portant  to  notice  that  Einstein  does  not  seek  to 
build  up  a  model  to  explain  gravitation  but  merely 
proposes  a  theory  of  motions.  His  equations  describe 
the  motion  of  any  body  in  terms  of  co-ordinates  of 
the  space-time  manifold,  making  use  of  the  inter- 
changeability  and  equivalence  implied  in  relativity. 
He  does  not  discuss  forces  as  such  ;  they  are,  after 
all,  as  Karl  Pearson  states,  "  arbitrary  conceptual 
measures  of  motion  without  any  perceptual  equiva- 
lent." They  are  simply  intermediaries  which  have 
been  inserted  between  matter  and  motion  from 
analogy  with  our  muscular  sense. 


THE  FOUNDATIONS  OF  EINSTEIN'S 

A  direct  consequence  of  the  application  of  the 
Principle  of  Equivalence  in  its  general  form  is  that 
the  velocity  of  light  varies  for  different  gravitational 
fields,  and  is  constant  only  for  uniform  fields  (this 
does  not  contradict  the  special  theory  of  relativity, 
which  was  built  up  for  uniform  fields,  and  only 
makes  it  a  special  case  of  this  much  more  general 
theory  of  relativity) .  But  change  of  velocity  implies 
refraction,  i.e.  a  ray  of  light  must  have  a  curved 
path  in  passing  through  a  variable  field  of  gravita- 
tion. This  affords  a  very  valuable  test  of  the  truth 
of  the  theory,  since  a  star,  the  rays  from  which  pass 
very  near  the  sun  before  reaching  us,  would  have  to 
appear  displaced  (owing  to  the  stronger  gravitational 
field  around  the  sun),  in  comparison  with  its  relative 
position  when  the  sun  is  in  another  part  of  the 
heavens  :  this  effect  can  only  be  investigated  during 
a  total  eclipse  of  the  sun,  when  its  light  does  not 
overpower  the  rays  passing  close  to  it  from  the  star 
in  question.*  The  calculated  curvature  is,  of  course, 
exceedingly  small  (1-7  seconds  of  arc),  but,  never- 
theless, should  be  observable. 

The  motion  of  the  perihelion  of  Mercury,  discovered 
by  Leverrier,  which  long  proved  an  insuperable 
obstacle  regarded  in  the  light  of  Newtonian 
mechanics,  is  immediately  accounted  for  by  the 
general  theory  of  relativity  ;  this  is  a  very  remark- 
able confirmation  of  the  theory. 

Before  we  finally  enunciate  the  general  theory  of 
relativity,  it  is  necessary  to  consider  a  special  form 
of  acceleration,  viz.  rotation.  Let  us  suppose  a 
space-time-domain  (referred  to  a  rigid  body  K)  in 
which  the  first  Newtonian  Law  holds,  i.e.  a  Galilean 
field  :  we  shall  suppose  a  second  rigid  body  of  refer- 
ence K1  to  be  rotating  uniformly  with  respect  to  K, 

*  We  shall  return  to  this  test  at  the  conclusion  of  the  chapter. 


THEORY  OF  GRAVITATION  125 

say  a  plane  disc  rotating  in  its  plane  with  constant 
angular  velocity.  An  observer  situated  on  the  disc 
near  its  periphery  will  experience  a  force  radially 
outwards,  which  is  interpreted  by  an  external 
observer  at  rest  relatively  to  K  as  centrifugal  force, 
due  to  the  inertia  of  the  rotating  observer.  But 
according  to  the  principle  of  equivalence  the  rotating 
observer  is  justified  in  assuming  himself  to  be  at  rest, 
i.e.  the  disc  to  be  at  rest.  He  regards  the  force 
acting  on  him  as  an  effect  of  a  particular  sort  of 
gravitational  field  (in  which  the  field  vanishes  at  the 
centre  and  increases  as  the  distance  from  the  centre 
outwards).  This  rotating  observer,  who  considers 
himself  at  rest,  now  performs  experiments  with 
clocks  and  measuring-scales  in  order  to  be  able  to 
define  time-  and  space-data  with  reference  to  K1. 
It  is  easy  to  show  that  if,  of  two  clocks  which  go  at 
exactly  the  same  rate  when  relatively  at  rest  in  the 
Galilean  field  K,  one  be  placed  at  the  centre  of  the 
rotating  disc  and  one  at  the  circumference,  the  latter 
will  continually  lose  time  as  compared  with  the 
former. 

Secondly,  if  an  observer  at  rest  in  K  measure  the 
radius  and  circumference  of  the  rotating  disc,  he  will 
obtain  the  same  value  for  the  radius  as  when  the 
disc  is  at  rest,  but  since,  when  he  measures  the  cir- 
cumference of  the  disc,  the  scale  lies  along  the 
direction  of  motion,  it  suffers  contraction,  and, 
consequently,  will  divide  more  often  into  the  circum- 
ference than  if  the  scale  and  the  disc  were  at  rest. 
(The  circumference  does  not  change,  of  course,  in 
rotation.)  That  is,  he  would  get  a  value  greater 

,,  r      ,,         ,.    circumference.     „„  .  ,, 

than  77  for  the  ratio  — ^ —  This  means  that 

diameter 

Euclidean  geometry  does  not  hold  for  the  observer 
making  his  observations  on  the  disc,  and  we  are 
obliged  to  use  co-ordinates  which  will  enable  his 


126      THE  FOUNDATIONS  OF  EINSTEIN'S 

results  to  be  expressed  consistently.  Gauss  in- 
vented a  method  for  the  mathematical  treatment  of 
any  continua  whatsoever,  in  which  measure-relations 
("  distance "  of  neighbouring  points)  are  denned. 
Just  as  many  numbers  (Gaussian  or  curvilinear 
co-ordinates)  are  assigned  to  each  point  as  the 
continuum  has  dimensions.  The  allocation  of  num- 
bers is  such  that  the  uniqueness  of  each  point  is 
preserved  and  that  numbers  whose  difference  is 
infinitely  small  are  assigned  to  infinitely  near  points. 
This  Gaussian  or  curvilinear  system  of  co-ordinates 
is  a  logical  generalization  of  the  Cartesian  system. 
It  has  the  great  advantage  of  also  being  applicable 
to  non-Euclidean  continua,  but  only  in  the  cases  in 
which  infinitesimal  portions  of  the  continuum  con- 
sidered are  of  the  Euclidean  form.  This  calls  to  mind 
the  remarks  made  at  the  commencement  of  this 
sketch  about  the  validity  of  geometrical  theorems. 
It  seems  as  though  the  miniature  view  that  we 
can  take  of  straight  lines  in  the  immensity  of  space 
led  to  a  firm  belief  in  the  universal  significance 
of  Euclidean  geometry.  When  we  deal  with  light 
phenomena  which  range  to  enormous  distances,  we 
find  that  we  are  not  justified  in  confining  ourselves 
to  Euclidean  geometry  ;  the  "  straightest  "  line  in 
the  time-space-manifold  is  "  curved."  We  must 
therefore  choose  that  geometry  which,  expressed 
analytically,  enables  us  to  describe  observed  pheno- 
mena most  simply  :  it  is  clear  that  for  even  large 
finite  portions  of  space  the  non-Euclidean  geometry 
chosen  must  practically  coincide  with  Euclidean 
geometry. 

We  now  see  that  the  general  theory  of  relativity 
cannot  admit  that  all  rigid  bodies  of  reference  K,  K1, 
etc.,  are  equally  justifiable  for  the  description  of  the 
general  laws  underlying  the  phenomena  of  physical 
nature,  since  it  is,  in  general,  not  possible  to  make 


THEORY  OF  GRAVITATION  127 

use  of  rigid  bodies  of  reference  for  space-time  descrip- 
tions of  events  in  the  manner  of  the  special  theory  of 
relativity.  Using  Gaussian  co-ordinates,  i.e.  label- 
ling each  point  in  space  with  four  arbitrary  numbers 
in  the  way  specified  above  (three  of  these  correspond 
to  three  space  dimensions  and  one  to  time),  the 
general  principle  of  relativity  may  be  enunciated 
thus  :— 

All  Gaussian  four-dimensional  co-ordinate-sys- 
tems are  equally  applicable  for  formulating  the 
general  laws  of  physics.  This  carries  the  principle 
of  relativity,  i.e.  of  equivalence  of  systems,  to  an 
extreme  limit. 

With  regard  to  the  relativity  of  rotations,  it  may 
be  briefly  mentioned  that  centrifugal  forces  can, 
according  to  the  general  theory  of  relativity,  be  due 
only  to  the  presence  of  other  bodies.  This  will  be 
better  understood  by  imagining  an  isolated  body 
poised  in  space  ;  there  could  be  no  meaning  in  saying 
that  it  rotated,  for  there  would  be  nothing  to  which 
such  a  rotation  could  be  referred  :  classical  mechanics 
however,  asserts  that,  in  spite  of  the  absence  of  other 
bodies,  centrifugal  forces  would  manifest  themselves  : 
this  is  denied  by  the  general  theory  of  relativity. 
No  experimental  test  has  hitherto  been  devised 
which  could  be  carried  out  practically  to  give  a 
decision  in  favour  of  either  theory. 

A  favourable  opportunity  for  detecting  the  slight 
curvature  of  light  rays  (which  is  predicted  by  the 
general  theory)  when  passing  in  close  vicinity  to  the 
sun  occurred  during  the  total  eclipse  of  the  2Qth  May, 
1919.  The  results,  which  were  made  public  at  the 
meeting  of  the  Royal  Society  on  6th  November 
following,  were  reported  as  confirming  the  theory. 

In  addition  to  the  slight  motion  of  Mercury's  peri- 
helion, there  is  still  a  third  test  which  is  based  upon 
a  shift  of  the  spectral  line  towards  the  infra-red,  as 


128     THE  FOUNDATIONS  OF  EINSTEIN'S 

a  result  of  an  application  of  Doppler's  principle  ; 
this  has  not  yet  led  to  a  conclusive  experimental 
result. 


I.  NOTE  ON  NON-EUCLIDEAN  GEOMETRY 

In  practical  geometry  we  do  not  actually  deal  with 
straight  lines,  but  only  with  distances,  i.e.  with 
finite  parts  of  straight  lines,  yet  we  feel  irresistibly 
impelled  to  form  some  conception  of  the  parts  of  a 
straight  line  which  vanish  into  inconceivably  distant 
regions.  We  are  accustomed  to  imagining  that  a 
straight  line  may  be  produced  to  an  infinite  distance 
in  either  direction,  yet  in  our  mathematical  reasoning 
we  find  that  in  order  to  preserve  consistency  (in 
Euclid),*  we  may  only  allocate  to  this  straight  line 
one  point  at  infinity  :  we  say  that  two  straight  lines 
are  parallel  when  they  cut  at  a  point  at  infinity, 
i.e.  this  point  is  at  an  infinite  distance  from  an 
arbitrary  starting-point  on  either  straight  line,  and  is 
reached  by  moving  forwards  or  backwards  on  either. 

Many  attempts  have  been  made,  without  success, 
to  deduce  Euclid's  "  axiom  of  parallels,"  which 
asserts  that  only  one  straight  line  can  be  drawn 
parallel  to  another  straight  line  through  a  point 
outside  the  latter,  from  the  other  axioms.  It  finally 
came  to  be  recognized  that  this  axiom  of  parallels 
was  an  unnecessary  assumption,  and  that  one  could 
quite  well  build  up  other  geometries  by  making  other 
equally  justified  assumptions. 

If  we  consider  a  point,  P,  outside  a  straight  line,  L 
(Fig.  4),  to  send  out  rays  in  all  directions,  then, 
starting  from  the  perpendicular  position  Palt  we 
find  that  the  more  obliquely  the  ray  falls  on  L 
the  farther  does  the  point  of  intersection  an  travel 

*  According  to  the  modern  analytical  interpretation  of  Euclid. 


THEORY  OF  GRAVITATION 


129 


along  L  to  the  left  (say).  Our  experience  teaches 
us  that  the  ray  and  L  have  one  point  in  common. 
There  is  no  justifiable  reason,  however,  for  asserting, 
as  Euclid's  axiom  does,  that  for  a  final  infinitely 
small  increase  of  the  angle  at  Pan  (i.e.  additional  turn 
of  Pan  about  P),  an  suddenly  bounds  off  to  infinity 
along  L,  i.e.  an,  the  point  of  intersection  leaves  finite 
regions  to  disappear  into  so-called  "  infinity,"  and 
that,  for  a  further  infinitesimal  increase,  an,  reappears 
at  infinity  at  the  other  end  of  L  to  the  right  of  04. 

One  might  equally  well  assume,  as  Lobatschewsky 
did,  that  P^QQ  and  Pa  _  ^  form  an  angle  which  differs 
ever  so  slightly  from  two  right  angles,  and  that  there 
are  an  infinite  number  of  other  straight  lines  included 


OC4    CX3   OC2   OC, 
FIG.  4. 


between  these  two  positions  (as  indicated  by  the 
dotted  lines  in  the  figure),  which  do  not  cut  L  at 
all.  Lobatschewsky  (andalso  Bolyai)  built  up  an 
entirely  consistent  geometry  on  this  latter  assump- 
tion. 

Riemann  later  abolished  the  assumption  of  infinite 
length  of  a  straight  line,  and  assumed  that  in  travel- 
ling along  a  straight  line  sufficiently  far  one  finally 
arrives  at  the  starting-point  again  without  having 
encountered  any  limit  or  barrier.  This  means  that 
our  space  is  regarded  as  being  finite  but  unbounded.* 

*  E.g.  the  surface  of  a  sphere  cuts  a  finite  volume  out  of  space, 
but  particles  sliding  on  the  surface  nowhere  encounter  boundaries 
or  barriers.  This  is  a  three-dimensional  analogon  to  the  four- 


130     THE  FOUNDATIONS  OF  EINSTEIN  S 

Thus  in  Riemann's  case  there  is  no  parallel  line  to 
L  for  an  never  leaves  L  ;  there  is  no  a^  .  This 
geometry  was  called  by  Klein  elliptical  geometry 
(and  includes  spherical  geometry  as  a  special  case). 
He  calls  Euclidean  geometry  parabolic  (Fig.  5),  for 
the  branches  of  a  parabola  continue  to  recede  from 
one  to  another,  and  yet  in  order  to  obtain  consistent 
results  in  its  formulae  we  are  obliged  only  to  assign 
one  point  at  infinity  to  it,  just  as  to  the  Euclidean 
straight  line.  Lobatschewsky's  geometry  is  simi- 
larly called  hyperbolic  (Fig.  5),  since  a  hyperbola  has 


FIG.  5. 

two  points  at  infinity,  corresponding  in  analogy  to 
the  two*points  at  infinity  at  which  the  two  parallels 
through  a  point  external  to  a  straight  line  cut  the 
latter. 

The  fact  that  one  is  obliged  to  renounce  Euclidean 
geometry  in  the  general  theory  of  relativity  leads  to 
the  conclusion  that  our  space  is  to  be  regarded  as 
finite  but  unbounded  :  it  is  curved,  as  Einstein 
expresses  it,  like  the  faintest  of  ripples  on  a  surface 
of  water. 

dimensional  space-time  manifold  of  Minkowski.  It  does  not  mean 
that  the  universe  is  enclosed  by  a  spherical  shell,  as  was  supposed  by 
the  ancients.  We  cannot  form  a  picture  of  the  corresponding  result 
in  the  four-dimensional  continuum  in  which,  according  to  the  general 
theory  of  relativity,  we  live. 


THEORY  OF  GRAVITATION  131 


SOME  ASPECTS  OF  RELATIVITY 

THE  THIRD  TEST 
BY  HENRY  L.  BROSE,  M.A. 

UP  to  the  present,  three  methods  of  verifying 
Einstein's  Theory  of  Relativity  have  been 
suggested. 

The  first  one,  which  was  a  direct  outcome  of  the 
new  gravitational  field-equations  proposed  by  Ein- 
stein, proved  successful.  The  slow  motion  of  Mer- 
cury's perihelion  which  long  mystified  astronomers 
was  immediately  accounted  for.  This  result  is  the 
more  remarkable  as  all  other  explanations  of  this 
phenomena  were  artificial  in  origin,  consisting  of  a 
hypothesis  formulated  ad  hoc  which  could  not  be 
verified  by  observation. 

The  second  method  involved  the  deflection  of  a 
ray  of  light  in  its  passage  through  a  varying  gravita- 
tional field.  The  results  of  the  total  eclipse  of  the 
sun  which  occurred  on  2Qth  May,  1919,  have  become 
famous  and  were  recorded  as  confirming  Einstein's 
prediction.  The  results  of  a  more  recent  expedition 
have  proved  finally  conclusive. 

The  third  test,  the  results  of  which  are  still  in 
abeyance,  is  perhaps  the  most  important  of  the  three, 
inasmuch  as  it  depends  upon  a  very  simple  calcula- 
tion from  Einstein's  Principle  of  Equivalence,  which 
asserts  that  an  observer  cannot  discriminate  how 


1S2      THE  FOUNDATIONS  OF  EINSTEIN'S 

much  of  his  motion  is  due  to  a  gravitational  field  and 
how  much  is  due  to  an  acceleration  of  his  body  of 
reference.  Einstein  illustrates  his  argument  by 
supposing  an  observer  situated  in  a  closed  box  in 
free  space.  The  observer  has  at  first  no  sensation 
of  weight,  and  need  not  support  himself  upon  his 
feet.  Now  suppose  an  external  agent  to  pull  the 
box  in  a  definite  direction  with  constant  force.  The 
observer  in  the  box  performs  experiments  with 
masses  of  variable  material,  and  as  they  all  fall  to 
the  "  floor  "  of  the  box  at  the  same  time,  he  concludes 
that  he  is  in  a  gravitational  field.  He  himself  has 
acquired  the  sensation  of  weight.  This  result  led 
Einstein  to  propound  the  equivalence  of  gravitational 
and  accelerational  fields.  An  immediate  consequence 
of  this  principle  is  that  the  duration  of  an  event 
depends  upon  the  gravitational  conditions  at  the 
place  of  the  event. 

If  we  consider  the  light  (of  frequency  j/J  which  is 
emitted  by  a  distant  star,  and  suppose  it  to  traverse 
a  practically  invariable  gravitational  field  in  which 
bodies  are  assumed  to  fall  with  a  constant  accelera- 
tion g,  then  an  observer  at  a  distance  h  from  the 
star  will  have  attained  a  velocity  (=  acceleration  x 

time)  =  g  .  -  where  c  is  the  velocity  of  light  and  the 

distance  h  is  small  in  comparison  with  the  distance 
traversed  by  the  observer  in  the  time  the  light  takes 
to  reach  him.  By  Doppler's  Principle,  the  apparent 
frequency  j>2  is  given  by 


='.(•+  3  = 


Potential  of  unit  mass  moved  through  a  distance 
h  is  gh  =  +  (f>  (say).  This  gives  the  work  done  in 
moving  unit  mass  from  the  source  of  light  to  the 


THEORY  OF  GRAVITATION  133 

observer  (the  source  of  light  is  here  the  point  to 
which  the  potential  energy  is  referred  in  the  field)  . 

Therefore,  if  we  transform  the  accelerational  field 
of  the  observer  into  the  gravitational  field,  we  get 
the  result  : 


This  means  that  a  spectral  line  of  frequency  vl 
will  appear  to  a  distant  observer  to  be  displaced,  if 
compared  with  the  position  of  the  same  line,  when 
produced  by  a  source  at  a  different  point  in  the 
field.  Each  of  these  lines,  produced  by  vibrat- 
ing electrons,  may  be  regarded  as  a  clock,  and  this 
simple  calculation  shows  how  time-measurements 
are  affected  by  the  state  of  the  gravitational  field. 
This  effect  amounts  to  0*008  Angstroms,  for  a  wave- 
length of  4000  A.  The  same  displacement  would  be 
produced  as  a  Doppler  effect  by  a  velocity  of  0-6  kms. 
per  sec.  When  this  test  was  put  into  practice,  it 
was  found  difficult  to  discriminate  it  from  the 
various  superposed  effects  due  to  other  causes  such 
as  the  radial  velocities  of  the  stars,  proper  velocities 
of  the  gaseous  envelopes,  pressure,  etc.  The  con- 
ditions of  the  emission  of  light  by  the  sun  have  not 
been  fully  ascertained,  nor  is  the  light  of  the  arc 
lamp  free  from  disturbing  elements.  Dr.  Erwin 
Freundlich,  of  the  Neubabelsberg  Observatory,  has 
discussed,  in  conjunction  with  Professor  Einstein, 
the  possibility  of  recognizing  this  effect  in  spite  of 
these  obscuring  influences.  He  points  out  three 
ways  of  establishing  the  result  qualitatively.  They 
may  be  briefly  classified  as  being  based  on  (i)  statis- 
tical methods  ;  (2)  nebular  spectra  ;  (3)  calcium 
lines  in  the  spectra  of  the  atmosphere  surrounding 
double-stars. 

i.  If  we  consider  a  great  number  of  stars  of  about 


134     THE  FOUNDATIONS  OF  EINSTEIN'S 

the  same  mass  evenly  distributed  over  the  heavens, 
and  represent  the  spectral  shift  due  to  radial  veloci- 
ties (i.e.  velocities  in  the  line  of  sight)  graphically, 
we  should  expect  these  velocities  to  be  distributed 
according  to  the  law  of  probability  about  the  value 
zero,  i.e.  as  depicted  by  Gauss's  Error  Curve,  which 
resembles  a  vertical  section  of  a  bell.  If,  however, 
Einstein's  gravitational  effect  really  exists,  we  should 
expect  these  velocities  to  group  themselves  sym- 
metrically about  a  positive  velocity  which  would  be 
that  corresponding  to  this  spectral  shift.  Gauss's 
Error  Curve  would  thus  appear  displaced  by  pre- 
cisely the  amount  of  the  radial  velocity  corresponding 
to  this  shift,  as  all  the  radial  velocities  would  be 
falsified  by  just  this  amount. 

The  values  of  the  radial  velocities  have  been 
plotted  in  the  case  of  B-stars,  called  Helium  stars 
on  account  of  the  predominance  of  helium  lines  in 
their  spectra.  Other  observations  have  led  astro- 
nomers to  infer  that  the  B-stars  have  unusually 
great  masses  but  small  densities.  The  result  has 
been  distinctly  in  favour  of  the  Einstein  shift  on 
the  basis  of  the  foregoing  discussion.  The  same  was 
found  to  hold  for  the  bright  K-  and  M-stars,  which 
are  considered  to  be  at  a  lower  temperature  and 
possessed  of  enormous  surface  extent,  which  ac- 
counts for  their  brilliance. 

If  we  indicate  the  mean  shift  of  the  lines  towards 
the  red  by  K,  then  for 

B-stars  K  =  4-3  kms.  +  0-5, 
K  „  K  =  3-5  „  ±0-9, 
M  „  K  =  5-3  „  +2-3. 

K  is  here  expressed  in  terms  of  a  Doppler  shift  as  a 
velocity,  i.e.  as  if  the  Einstein  shift  were  due  to  an 
additional  radial  motion  and  hence  expressible  in 
kilometres. 


THEORY  OF  GRAVITATION  135 

Alternative  ways  of  accounting  for  this  shift  have 
been  proposed. 

(a)  It  may  be  regarded  as  an  ordinary  Doppler 
effect.  This  would  imply  that  the  stars  of  the  B, 
K,  and  M  type  suffer  a  general  expansion  to  which 
stars  of  the  F  and  G  type  (yellow  stars  like  the  sun) 
and  the  A-stars  are  not  subject. 

This  explanation  does  not  seem  very  probable,  as 
helium  lines  were  used  in  determining  the  shift  for 
the  B-stars,  whereas  quite  different  lines  were  used 
for  measuring  the  effect  for  the  K-  and  M-stars. 
It  would  be  a  strange  coincidence  if  this  shift,  to 
which  all  the  evidence  points  as  arising  from  a 
common  origin,  should  be  manifested  just  in  these 
cases  which  have  been  made  the  object  of  an  investi- 
gation. 

(6)  The  general  shift  towards  the  red  might  be 
ascribed  to  pressure  effects  at  the  surfaces  of  the 
stars  or  to  the  presence  of  other  lines  which  lie  on 
the  red  side  of  the  main  lines,  but  which  are  very 
weak  or  even  absent  in  the  comparison  spectrum  of 
the  sun.  A  detailed  knowledge  of  conditions  on  the 
surfaces  of  stellar  bodies  could  alone  give  a  decision 
on  this  point. 

2.  It  is  only  possible  to  prove  that  the  shift  K 
is  not  due  to  a  radial  velocity  if  one  can  measure 
the  ordinary  Doppler  effect  arising  from  the  radial 
velocity  separately.  Let  us  consider  a  single  B-star 
or  group  of  B-stars  which  happen  to  be  embedded 
in  a  nebula  of  great  extent  which  accompanies  them 
in  their  motion.  The  Doppler  effect  due  to  the 
radial  velocity  would  be  the  same  for  the  star  as  the 
nebula,  but  the  gravitational  effect  predicted  by 
Einstein  would  not  be  the  same,  inasmuch  as  the 
gravitational  field  at  the  surface  of  the  star  will  vary 
considerably  from  that  at  the  outer  edge  of  the 
nebula.  Hence  it  would  be  reasonable  to  attribute 


136     THE  FOUNDATIONS  OF  EINSTEIN'S 

any  difference  in  the  magnitude  of  the  spectral 
shifts  in  the  case  of  the  star  and  the  nebula  to  the 
difference  in  gravitational  fields  at  each  place. 

The  stars  of  the  nebular  group  of  Orion  have 
hitherto  offered  the  only  possibility  of  applying  this 
method.  The  results  have  fulfilled  Einstein's  expec- 
tations qualitatively,  and  it  remains  to  be  seen 
whether  the  agreement  will  hold  quantitatively.  A 
general  shift  of  the  star-spectrum  as  compared  with 
the  corresponding  lines  of  the  associated  nebula  was 
observed. 

Some  very  bright  B-stars  in  the  constellation  of 
Orion  are  considered  to  form  an  entity  with  their 
attendant  nebula.  This  conclusion  was  reached  as 
the  result  of  independent  research. 

The  radial  velocity  of  the  Orion-nebula  has  been 
measured  by  various  observers.  The  values  obtained 
are:  17-7  (Wright),  17-4  (Vogel  and  Eberhardt), 
18-5  (Frost  and  Adams).  The  mean  value  is  17-4 
kms.  per  sec.  This  velocity  is  derived  from  the 
brightest  part  of  the  nebula,  the  so-called  trapezium. 
The  values  obtained  in  the  case  of  the  stars  almost 
all  exceed  20  kms.  per  sec.,  and  hence  it  seems  likely 
that  part  of  this  radial  velocity,  viz.  the  excess  over 
that  of  the  nebula,  is  due  to  the  Einstein  effect. 
When  the  difference  between  the  radial  velocities  of 
the  stars  and  the  associated  nebula  are  tabulated  for 
each  star,  we  find  that  in  the  case  of  all  members 
except  two  the  difference  is  positive,  i.e.  indicative 
of  a  shift  towards  the  red  end,  in  agreement  with  the 
statistical  investigation  applied  to  the  B-,  K-,  and 
M-stars.  The  difference  amounts  to  6-0  kms. 
+  -i  km.,  and  is  a  little  greater  than  that  given  by 
the  statistical  method. 

The  two  stars  0  and  36  Orionis  give  a  displace- 
ment towards  the  violet  end.  It  has  been  suggested 
that  they  do  not  belong  to  the  more  limited  group  of 


THEORY  OF  GRAVITATION  137 

Orion  stars,  but  are  only  projected  into  that  portion 
of  the  celestial  sphere.  This  is  supported  by  the 
fact  that  both  stars  have  only  very  small  spherical 
proper  motions,  and  that  the  radial  velocities 
observed  for  them  differ  considerably  from  the  mean 
of  the  radial  velocities  of  the  others. 

This  method  has  not  been  successfully  applied  to 
other  stellar  systems  inasmuch  as  the  nebulae  of 
those  which  are  available  emit  such  feeble  light  that 
it  has  not  been  possible  to  establish  the  displacement 
to  any  degree  of  accuracy.  Eddington  recently 
pointed  out  that  a  very  important  factor  had  been 
neglected  in  the  fundamental  equations  of  the  early 
theories  concerning  the  equilibrium  of  stellar  matter, 
viz.  the  pressure  due  to  radiation.  According  to 
his  theory,  the  equilibrium  in  the  interior  of  the  star 
(regarded  as  a  gaseous  sphere)  is  determined  by  three 
conditions.  These  are  gaseous  pressure,  radiational 
pressure,  and  gravitational  forces. 

Calculation  shows  that  for  very  great  masses  the 
gravitational  pressure  is  almost  entirely  balanced  by 
radiational  pressure.  This  implies  that  any  addi- 
tional force  such  as  that  due  to  a  centrifugal  field  of 
rotation  would  lead  to  an  unstable  condition. 

It  can,  furthermore,  be  deduced  from  Eddington's 
theory  that  only  stars  whose  masses  exceed  a  certain 
minimum  value  can  in  the  course  of  their  evolution 
reach  the  very  high  surface-temperatures  which  have 
been  observed  in  the  case  of  the  O-  and  B-stars. 

It  therefore  seems  likely  that  the  O-  and  B-stars 
have  in  the  process  of  evolution  passed  through  a 
stage  of  which  the  radiational  pressure  has  brought 
about  a  condition  of  unstable  equilibrium,  and  one 
might  expect  them  to  be  surrounded  by  cosmic  dust 
which  has  become  dissociated  from  the  nuclei  of  the 
system. 

In  some  cases  this  dissociated  matter  may  be  in 
9* 


138     THE  FOUNDATIONS  OF  EINSTEIN'S 

a  very  fine  state  of  division,  and  may  extend  so  far 
into  space  that  the  absorption  lines  they  produce  in 
the  spectrum  of  the  star  they  surround  may  originate 
from  a  gravitational  field  which  differs  perceptibly 
from  that  at  the  surface  of  the  star.  There  are 
definite  signs  of  the  existence  of  such  atmospheres. 
A  high  percentage  of  B-stars  are  found  to  be  spectro- 
scopic  double  stars,  i.e.  their  spectral  lines  fluctuate 
periodically  about  some  mean  position.  Hartmann 
was  the  first  to  notice  that  in  the  spectrum  of  the 
B-star  S.  Orionis  the  absorption  lines  K  and  H  of 
calcium,  viz.  3933-82  and  3968-63  Angstroms,  occur, 
but  that  they  do  not  share  in  the  periodic  movements 
of  the  other  lines.  A  number  of  other  stars  belonging 
to  early  spectral  types  contain  calcium  absorption 
lines  in  their  spectra,  which  exhibit  a  similar  anomaly, 
inasmuch  as  they  either  remain  immovable  or  execute 
periodic  motions  which  are  of  feeble  amplitude  com- 
pared with  the  proper  stellar  lines.  In  view  of  the 
important  rdle  that  calcium  plays  in  the  outermost 
layers  of  the  gaseous  atmosphere  encircling  the  sun, 
and  in  view  of  the  discussion  above,  the  suggestion 
forces  itself  upon  one  that  these  calcium  lines  indicate 
the  presence  of  an  extensive  atmosphere  surrounding 
the  star. 

It  has  often  been  put  forward  that  these  lines  are 
due  to  the  light  from  these  stars  being  absorbed  by 
vast  interstellar  clouds  of  calcium.  Evershed  con- 
siders that  this  is  supported  by  the  fact  that  when 
the  motion  of  the  solar  system  is  subtracted  from 
that  calculated  from  the  fixed  calcium  lines  (owing 
to  the  ordinary  Doppler  effect),  the  remaining  motion 
is  very  small.  But  this  argument  does  not  carry 
weight  inasmuch  as  it  is  known  that  the  B-stars,  in 
the  spectrum  of  which  these  lines  occur,  themselves 
have  very  small  radial  velocities.  As  Young  re- 
marked, it  seems  very  strange  that  these  calcium 


THEORY  OF  GRAVITATION  139 

clouds  should  so  consistently  choose  to  lie  in  front  of 
stars  of  type  B  or  earlier.  An  objection  against  this 
hypothesis  is  to  be  found  in  the  fact  that  in  the 
case  of  various  systems  these  two  calcium  lines  are 
not  at  rest  but  move,  although  with  somewhat  less 
amplitude  than  the  other  proper  lines  of  the  double 
star. 

An  additional  circumstance  which  lends  support 
to  the  theory  that  calcium  lines  denote  the  presence 
of  an  atmosphere  around  the  star  is  that  a  great 
number  of  helium-stars  are  enveloped  in  a  nebulous 
atmosphere  which  is  actually  visible. 

Assuming  then  that  the  calcium  absorption  lines 
are  due  to  such  atmospheres,  we  may  apply  the  same 
process  as  in  the  case  of  the  Orion  nebula,  i.e.  if  the 
shifts  of  the  spectral-lines  of  the  stars  be  systematic- 
ally falsified  by  a  superposed  gravitational  effect, 
this  should  be  expressed  by  the  lines  of  the  actual 
spectrum  from  a  double  star  being  displaced  towards 
the  red  as  compared  with  the  fixed  calcium  lines. 

This  phenomenon  has  been  clearly  observed.  The 
result  has  not  yet  been  quantitatively  fixed,  as  the 
numbers  taken  are  not  regarded  as  final. 

All  stars  in  the  spectra  of  which  the  H  and  K  lines 
of  calcium  occur  have  been  used  to  test  the  con- 
clusion, and  all  show  a  shift  to  the  red  end ;  the 
mean  of  the  shifts  corresponds  to  a  velocity  of 
-j-  6-3  kms.  per  sec. 

The  results  of  this  discussion  have  been  formulated 
by  Dr.  Freundlich  thus  : — 

SUMMARY 

i.  Statistical  consideration  gives  us  the  means  of 
separating  the  mean  gravitational  effect  from  the 
ordinary  Doppler  effect  in  the  case  of  the  helium 
B-stars  and  the  bright  K-  and  M-stars,  which 


140    EINSTEIN'S  THEORY  OF  GRAVITATION 

astronomical  investigations  compel  us  to  regard  as 
being  of  particularly  great  mass. 

A  general  shift  of  the  spectra  towards  the  red  is 
exhibited  with  considerable  certainty. 

2.  It  follows  from  a  comparison  of  the  displacement 
of  the  lines  of  the  star-spectra  that  the  above  dis- 
placement which  was  found  by  a  statistical  examina- 
tion is  not  an  ordinary  Doppler  effect,  but  is  due  to 
the  conditions  of  emission  of  light  at  the  surfaces  of 
the  stars. 

3.  The  close  connection  of  the  B-  and  0-stars  with 
nebulous  matter  in  the  heavens  is  a  symptom  that 
these  stars  are  of  great  mass. 

4.  If  we  regard  the  fixed  calcium  lines  in  the 
spectra  of  B-  and  0-stars  as  being  caused  by  absorp- 
tion in  extended  calcium  atmospheres  moving  with 
each  star  in  question,  the  shift  towards  the  red 
which  manifests  itself  may  be  regarded  as  the  effect 
predicted   by   Einstein's   theory,   i.e.    due    to    the 
different  gravitational  fields  from  which  the  absorp- 
tion lines  and  the  stellar  lines  have  originated. 


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